Lattice-based equation of state at finite baryon number, electric charge and strangeness chemical potentials
LLattice-based equation of state at finite baryon number, electric charge andstrangeness chemical potentials
J. Noronha-Hostler a,b , P. Parotto c,d , C. Ratti d , J. M. Stafford d a Department of Physics and Astronomy, Rutgers University, Piscataway, NJ USA 08854 b Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA c University of Wuppertal, Department of Physics, Wuppertal D-42097, Germany and d Department of Physics, University of Houston, Houston, TX, USA 77204 (Dated: January 23, 2020)We construct an equation of state for Quantum Chromodynamics (QCD) at finite temperatureand chemical potentials for baryon number B , electric charge Q and strangeness S . We use theTaylor expansion method, up to the fourth power for the chemical potentials. This requires theknowledge of all diagonal and non-diagonal BQS correlators up to fourth order: these resultsrecently became available from lattice QCD simulations, albeit only at a finite lattice spacing N t =12. We smoothly merge these results to the Hadron Resonance Gas (HRG) model, to be ableto reach temperatures as low as 30 MeV; in the high temperature regime, we impose a smoothapproach to the Stefan-Boltzmann limit. We provide a parameterization for each one of these BQS correlators as functions of the temperature. We then calculate pressure, energy density, entropydensity, baryonic, strangeness, electric charge densities and compare the two cases of strangenessneutrality and µ S = µ Q = 0. Finally, we calculate the isentropic trajectories and the speed ofsound, and compare them in the two cases. Our equation of state can be readily used as an inputof hydrodynamical simulations of matter created at the Relativistic Heavy Ion Collider (RHIC). INTRODUCTION
Relativistic heavy ion collisions have successfully recre-ated the Quark Gluon Plasma (QGP) in the labora-tory at the Relativistic Heavy Ion Collider (RHIC) atBrookhaven National Laboratory and the Large HadronCollider (LHC) at CERN. At low baryon densities, thetransition from the hadron gas phase where quarks andgluons are confined within hadrons into a deconfinedstate where quark and gluons are the main degrees of free-dom is a smooth cross-over [1–3]. At larger baryon densi-ties, the phase transition is expected to become stronger,eventually turning into first-order. If this is the case,there has to be a critical point on the QCD phase di-agram [4–8]. The search for the QCD critical point isthe focus of the second Beam Energy Scan (BES II) atRHIC, running in 2019 and 2020.The Quark Gluon Plasma acts as a nearly perfect fluidand as such can be well-described by event-by-event rel-ativistic viscous hydrodynamical models. The hydrody-namical description of the fireball has proved to be verysuccessful in describing the experimental data [9–19]. Inorder to close the hydrodynamical equations, an Equa-tion of State (EoS) is required, which is based on firstprinciple Lattice QCD calculations. Recently, a Bayesiananalysis [20] has provided an important validation of thelattice QCD equation of state. This framework, based ona comparison of data from the LHC to theoretical mod-els, has applied state-of-the-art statistical techniques tothe combined analysis of a large number of observableswhile varying the model parameters. The posterior dis-tribution over possible equations of states turned out to be consistent with results from lattice QCD simulations.Additionally, the correct description of the QCD equa-tion of state is needed because differences in the equationcan affect the extraction of transport coefficients [17].Thus, a lattice-based QCD equation of state is a funda-mental ingredient in the description of the state of mattercreated in a heavy-ion collision. The precise lattice QCDresults for several thermodynamic quantities can thus beused in support of the heavy ion experimental program[21].The EoS of QCD at zero baryonic density is knownwith high precision from first principles since a few years[22–24]. The calculation of the equation of state at finitechemical potential is hindered by the sign problem. Nev-ertheless, the thermodynamic quantities can be expandedas a Taylor series in powers of µ B /T , for which the co-efficients χ n can be simulated on the lattice at µ B = 0.From these Taylor coefficients a variety of Lattice QCDbased equations of state have been reconstructed [25–27]and later used within relativistic hydrodynamics [25, 28–31].However, baryon number is not the only conservedcharge in a heavy ion collision: strangeness and electriccharge are also relevant quantum numbers. In fact, manyquestions remain regarding a possible separate freeze-outtemperature for strange hadrons [32–35] and separationsof electric charge due to a possible chiral magnetic effect[36], so many interesting questions need to be answered,that go beyond just baryon charge conservation. At theLHC, where the baryonic chemical potential µ B is basi-cally vanishing, the chemical potentials for strangeness µ S and electric charge µ Q are also zero. At RHIC how- a r X i v : . [ h e p - ph ] J a n ever, as the baryonic density increases, the other twochemical potentials have finite values as well. Until now,the equation of state of QCD has only been extrapo-lated to finite µ B , either by keeping µ S = µ Q = 0, oralong a specific trajectory in the four-dimensional pa-rameter space, namely imposing that the strangenessdensity (cid:104) n S (cid:105) = 0 and that the electric charge density (cid:104) n Q (cid:105) = 0 . (cid:104) n B (cid:105) to match the experimental situation.After the early results for χ , χ and χ [37], a con-tinuum extrapolation for χ was published in Ref. [38];in Ref. [39] χ was shown, but only at finite lattice spac-ing. The continuum limit for χ was published for thefirst time in [40] in the case of strangeness neutrality, andlater in [41] for both cases. In [42], a first determinationof χ at two values of the temperature and N t = 8 waspresented. Finally, in Ref. [43] a determination of χ was presented for the first time as a function of the tem-perature, at N t = 12, keeping µ S = µ Q = 0. Recently,the effect of introducing a critical point in the equationof state of QCD has also been tested [26].However, a Taylor expansion of the equation ofstate, along a direction which satisfies the strangeness-neutrality condition is not enough for the hydrody-namics approach, since the fluid cells have local fluc-tuations in strangeness density. Additionally, there isa complicated interplay between transport coefficientswhen B, Q, S are considered [44] that cannot be ne-glected at large baryon densities. For these reasons,an EoS fully expanded as a Taylor series in powers of µ B /T, µ S /T, µ Q /T is needed as an input of hydrody-namic simulations of the matter created at RHIC. In or-der to perform such an expansion, all of the diagonaland non-diagonal susceptibilities of these three conservedcharges are needed from lattice QCD up to the chosenpower. In this work, we perform the Taylor expansion ofto total power four in the chemical potentials. These re-sults recently became available [43], on N t = 12 lattices.Alternative approaches to the Taylor series expansionhave been suggested in [45, 46] and [47, 48], which havebeen shown to match well to lattice QCD data for theFourier harmonics [49] at imaginary chemical potential.These Fourier harmonics appear to be important to dis-tinguish baryon interactions within a hadron resonancegas (see also [50]), specifically for the thermodynamicregime above T >
150 MeV. We note that here we uselattice QCD data entirely in this regime (our hadron res-onance gas model is only to constrain low temperaturesbelow T (cid:46)
135 MeV where no lattice QCD results areavailable). However, due to the Taylor expansion our ap- proach is limited to chemical potentials µ B (cid:46) (2 − . T .To fully reproduce the Fourier harmonics we would needto reach µ B (cid:46) πT , for which higher order coefficients inthe Taylor series would need to be included.In this manuscript, we construct an equation of statefor QCD at finite T , µ B , µ S , µ Q . We build the pres-sure as a Taylor series of the three chemical potentials,with coefficients taken from lattice simulations [43]. Atlow temperatures, we perform a smooth merging betweenthe lattice and the Hadron Resonance Gas model results[51] and ensure continuity of higher order derivatives. Athigh temperatures, we impose a smooth approach to theStefan-Boltzmann limit. We parameterize each one ofthese coefficients as a ratio of polynomials. From thiswe obtain the pressure and can then calculate all otherquantities from thermodynamic relationships . METHODOLOGY AND RESULTS
The Taylor series of the pressure in terms of the threeconserved charge chemical potentials can be written as p ( T, µ B , µ Q , µ S ) T = (cid:88) i,j,k i ! j ! k ! χ BQSijk (cid:16) µ B T (cid:17) i (cid:16) µ Q T (cid:17) j (cid:16) µ S T (cid:17) k . (1)We limit our calculation to i + j + k ≤
4. The coefficients χ BQSijk = ∂ i + j + k ( p/T ) ∂ ( µ B T ) i ∂ ( µ Q T ) j ∂ ( µ S T ) k (cid:12)(cid:12)(cid:12)(cid:12) µ B ,µ Q ,µ S =0 (2)have recently been published from lattice QCD simula-tions on 48 ×
12 lattices [43] in the temperature range(135 MeV) < T < (220 MeV). Since this is not enoughto cover the hydrodynamical evolution of the system,we smoothly merge each coefficient at low temperaturewith the Hadron Resonance Gas model result, while athigh temperature we calculate the Stefan-Boltzman limitfor each one of them and assume that their value at T = 800 MeV is ∼
10% away from the respective Stefan-Boltzmann limit. To simplify the notation, whenever i , j , k are zero, we only write the non-zero indices and onlythe corresponding conserved charges: for example, χ BQS becomes χ B , χ BQS becomes χ BS and so on. In order toprovide a smooth pressure which can be easily derived toobtain the other thermodynamic quantities, we parame-terize each coefficient by means of a ratio of up-to-ninthorder polynomials in the inverse temperature: Right before publishing this manuscript, we became aware ofRef. [52] which constructs a similar equation of state as the onepresented here. One major difference is that we match lattice QCD susceptibilities with the hadron resonance gas model beforereconstructing the equation of state whereas in [52] the matchingwith the HRG model is performed for the Taylor-reconstructedpressure. χ BQSijk ( T ) = a i + a i /t + a i /t + a i /t + a i /t + a i /t + a i /t + a i /t + a i /t + a i /t b i + b i /t + b i /t + b i /t + b i /t + b i /t + b i /t + b i /t + b i /t + b i /t + c . Only χ B requires a different parameterization: χ ( T ) = e − h /t (cid:48) − h /t (cid:48) · f · (1 + tanh( f t (cid:48) + f )) (3)In both equations above, t = T /
154 MeV, t (cid:48) = T /
200 MeV [53]. The values of the parameters for eachcoefficient are given in the appendix, together with the re-spective Stefan-Boltzmann limits. Figures 1 and 2 showall of the Taylor expansion coefficients as functions of thetemperature. The black dots are the HRG model results,the red triangles correspond to the lattice QCD resultsand the thick blue line indicates the Stefan-Boltzmannlimit.Making use of this parameterization, we construct thepressure from Eq. (1). The other thermodynamic quan-tities are then derived from the pressure as follows: sT = 1 T ∂p∂T (cid:12)(cid:12)(cid:12)(cid:12) µ i , (cid:15)T = sT − pT + (cid:88) i µ i T n i T n i T = 1 T ∂p∂µ i (cid:12)(cid:12)(cid:12)(cid:12) T,µ j , c s = ∂p∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) n i + (cid:88) i n i (cid:15) + p ∂p∂n i (cid:12)(cid:12)(cid:12)(cid:12) (cid:15),n j . (4)Everywhere in the above equation, i (cid:54) = j is intended.In Fig. 3 we show the dependence of the normal-ized pressure, entropy density, energy density, baryonic,strangeness and electric charge densities on the temper-ature, along lines of constant µ B /T = 0 . , ,
2, bothwith (cid:104) n S (cid:105) = 0, (cid:104) n Q (cid:105) = 0 . (cid:104) n B (cid:105) (solid black lines), andin the case µ S = µ Q = 0 (dashed red lines). We findthat the thermodynamic quantities that are less sensi-tive to the chemical composition of the system do notshow large discrepancies between the two scenarios, forall three values of µ B /T . On the other hand, when re-alistic conditions on the global chemical composition ofthe system are imposed, the baryon density is largely af-fected, and substantially decreased; the opposite effectis visible for the electric charge density, which is heavilyenhanced.Finally, we compare i) the isentropic trajectories, ii)the temperature dependence of the speed of sound alonglines of constant µ B /T and iii) the behavior of the speedof sound along parametrized chemical freeze-out lines be-tween these two cases. The isentropic trajectories areshown in Fig. 4 for selected values of s/n B , which cor-respond to the indicated collision energies [40]. In theupper panel of Fig. 5 we show the speed of sound asa function of the temperature along lines with µ B /T =0 . , ,
2; the different colors correspond to different val-ues of µ B /T . In the lower panel of Fig. 5 we show thebehavior of the speed of sound along two parametrizedchemical freeze-out lines. These two freeze-out lines are shifted from the one presented in [54], and have the form: T F O ( µ B ) = T + bµ B + cµ B , (5)with b = − . · − MeV − and c = − . · − MeV − ;the two lines we show have T F O ( µ B = 0) = 160 MeV and T F O ( µ B = 0) = 150 MeV. Both in Fig. 4 and in Fig. 5,the solid lines correspond to (cid:104) n S (cid:105) = 0, (cid:104) n Q (cid:105) = 0 . (cid:104) n B (cid:105) while the dashed lines to µ S = µ Q = 0.Since the EoS constructed in this work is a Taylorexpansion carried out from lattice-QCD-calculated ex-pansion coefficients, it is important to have an idea ofthe range of the validity of such expansion. It has beenshown from lattice QCD simulations that the Taylor ex-pansion of the Equation of State up to O ( µ B ) convergesfor µ B /T (cid:46) − . √ s (cid:38)
10 GeV [54]. In order to have a better idea of wherea possible breakdown of its validity occurs, we show inFig. 6 the behavior of the electric chemical potential inthe case with strangeness neutrality, along lines of con-stant µ B /T = 0 . −
3. We see that a non-monotonicbehavior appears around and above µ B /T ∼ .
5. This isin line with the expectation that the convergence of theTaylor series is guaranteed in the regime µ B /T (cid:46) . µ B – and thus reproduce theFourier harmonics from [49] – since for them the coverageof the region µ B /T ≤ π would be required. Applying theconstraints from imaginary µ B can be done in the nearfuture to further improve our modeling of the QCD EoS,possibly concurrently with the inclusion of new contin-uum extrapolated lattice results. CONCLUSIONS
In this manuscript, we constructed an equation of statefor QCD at finite temperature and
B, Q, S chemical po-tentials, based on a Taylor series up to fourth power inthe chemical potentials. Our methodology is based ona smooth merging between the HRG model and latticeQCD results for each one of the Taylor expansion coeffi-cients; for all coefficients except χ B , the parameterizationfunction is a ratio of up-to-ninth order polynomials. Weprovide all parameters in Tables I,II,III, so that our EoScan be readily used in the community. Furthermore, thecode to generate the EoS and the tables for the ther-modynamic quantities as functions of T, µ B , µ S , µ Q isavailable at the link mentioned in Ref. [55]. χ B lattice HRG Parameterization SB limit χ Q χ S -0.01 0 0.01 0.02 0.03 0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 χ B Q -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 χ BS χ Q S χ B T [GeV] 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 χ Q T [GeV] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 χ S T [GeV]
FIG. 1. From left to right, top to bottom: expansion coefficients χ B , χ Q , χ S , χ BQ , χ BS , χ QS , χ B , χ Q , χ S as functions ofthe temperature. In each panel, the black dots are the HRG model results, the red triangles correspond to the lattice QCDresults and the thicker blue line on the right indicates the Stefan-Boltzmann limit. The thin solid, black curve shows ourparameterization of the data. The Equation of State presented in this manuscript isimportant for the hydrodynamic description of the sys-tem created in heavy ion collisions at RHIC. There arenumerous outstanding questions that remain to be under-stood at finite baryon densities that are influenced bothby electric charge and strangeness. One recent surprisethat arose from the first Beam Energy Scan was Λ po-larization, that indicates that the Quark Gluon Plasmamay be the most vortical fluid known to humanity [56].However, considering that Λ’s are simultaneously bothstrange particles and baryons, polarization studies shouldbe done in hydrodynamic simulations that also considerall three conserved charges because of this interplay be-tween strangeness and baryon number. As previouslymentioned, this
BQS equation of state can help shedlight on the possible flavor hierarchy of freeze-out tem-peratures as well as the chiral magnetic effect. A varietyof dynamical observables of conserved charges (e.g. kaon flow harmonics) have already been measured at the BeamEnergy Scan I and many others are planned for the BeamEnergy Scan II, which may help to further constrain thelocation of a possible critical point.Finally, we point out that strange hadrons make uproughly 10% of all measured hadrons (assuming the kaonto pion ratio is a reasonable estimate for the ratio of allfinal state hadrons) and we can primarily only measurecharged particles . Thus, a BQS equation of state isrequired for a fully consistent description of the QuarkGluon Plasma at finite densities. Relativistic hydrody-namics in the presence of multiple conserved charges ob-tains cross terms that affect the transport coefficients[44, 57, 58]. Thus, it is misleading to extract transport Some neutral particles can be reconstructed from their daughterparticles e.g. π → γγ -0.01-0.005 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 χ B Q lattice HRG Parameterization SB limit -0.1-0.08-0.06-0.04-0.02 0 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 χ BS χ Q S -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 χ B Q -0.3-0.25-0.2-0.15-0.1-0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 χ BS χ Q S χ B Q χ BS χ Q S χ B Q S T[GeV] -0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 χ B Q S T [GeV] -0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 χ B Q S T [GeV]
FIG. 2. From left to right, top to bottom: expansion coefficients χ BQ , χ BS , χ QS , χ BQ , χ BS , χ QS , χ BQ , χ BS , χ QS , χ BQS , χ
BQS ,χ BQS , as functions of the temperature. In each panel, the black dots are the HRG model results, the red triangles correspondto the lattice QCD results and the thicker blue line on the right indicates the Stefan-Boltzmann limit. The thin solid, blackcurve shows our parameterization of the data. coefficients at finite baryon densities only considering fi-nite baryon number and not also finite strangeness andelectric charge. Furthermore, transport coefficients ofdifferent conserved charges have different characteristictemperatures, which further complicates the picture atlarge densities [59]. The consequences are still under de-velopment, but it is certain that a BQS equation of stateis a vital first step to take into account any of these ef- fects.At this point, our reconstructed
BQS equation of stateonly consists of a cross-over transition. Unlike a previouswork where an equation of state at finite µ B was coupledto the 3D Ising model in order to study criticality [26],such an endeavor with three conserved charges would besignificantly more complicated. While the term “criticalpoint” is used, there might actually be a critical line oreven critical plane once one considers the full three di-mensional space of µ B , µ S , and µ Q . Since there are largefluctuations in T , µ B , µ S , and µ Q throughout the evo-lution of a single event [60–62], certain elements of thefluid might pass through a critical region at an entirelydifferent combination of T , µ B , µ S , and µ Q . APPENDIX
We list the values of the parameters in Eq. (3) foreach Taylor expansion coefficient in Table I. The Stefan-Boltzmann limit for the coefficients have the following values: p ( T, , , T = 19 π , (6) χ B = 13 , χ Q = 23 , χ S = 1 ,χ BQ = 0 , χ BS = − , χ QS = 13 ,χ B = 29 π , χ Q = 43 π , χ S = 6 π ,χ BQ = 0 , χ BS = − π , χ QS = 29 π ,χ BQ = 49 π , χ BS = − π , χ QS = 2 π χ BQ = 49 π , χ BS = 23 π , χ QS = 23 π χ BQS = 29 π , χ BQS = − π , χ BQS = − π ACKNOWLEDGEMENTS
This material is based upon work supported by theNational Science Foundation under grants no. PHY-1654219 and OAC-1531814 and by the U.S. Departmentof Energy, Office of Science, Office of Nuclear Physics,within the framework of the Beam Energy Scan Theory(BEST) Topical Collaboration. We also acknowledge thesupport from the Center of Advanced Computing andData Systems at the University of Houston. J.N.H. ac-knowledges support from the US-DOE Nuclear ScienceGrant No. de-sc0019175. [1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K.Szabo, Nature , 675 (2006), arXiv:hep-lat/0611014[hep-lat].[2] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,C. Ratti, and K. K. Szabo (Wuppertal-Budapest), JHEP , 073 (2010), arXiv:1005.3508 [hep-lat].[3] A. Bazavov et al. , Phys. Rev. D85 , 054503 (2012),arXiv:1111.1710 [hep-lat].[4] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak,Phys. Rev. Lett. , 4816 (1998), arXiv:hep-ph/9806219[hep-ph].[5] A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A.Stephanov, and J. J. M. Verbaarschot, Phys. Rev. D58 ,096007 (1998), arXiv:hep-ph/9804290 [hep-ph].[6] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak,Phys. Rev.
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T, µ B ) plane, for s/n B = 420 , , ,
30, corresponding to collision ener-gies √ s NN = 200 , . , , . (cid:104) n S (cid:105) = 0, (cid:104) n Q (cid:105) = 0 . (cid:104) n B (cid:105) while thedashed red lines to µ S = µ Q = 0. μ B / T = μ B / T = μ B / T = μ B / T = μ B / T = μ B / T =
100 200 300 400 500 6000.100.150.200.250.300.35 T [ MeV ] c s T =
160 MeV T =
160 MeV SN T =
150 MeV T =
150 MeV SN μ B [ MeV ] c s FIG. 5. (Upper panel) Temperature dependence of the speedof sound along lines of constant µ B /T . The solid lines corre-spond to (cid:104) n S (cid:105) = 0, (cid:104) n Q (cid:105) = 0 . (cid:104) n B (cid:105) while the dashed ones to µ S = µ Q = 0. The curves for values of µ B /T = 0 . , , T = 450 MeV) and pink/lighter gray (this line stops at T = 225 MeV) respectively. (Lower panel) Behavior of thespeed of sound along parametrized chemical freeze-out linesas in Eq. (5), with T FO ( µ B = 0) = 160 MeV (pink/lightergray lines) and T FO ( µ B = 0) = 150 MeV (dark blue/darkergray lines). As in the upper panel, solid and dashed lines cor-respond to the cases with and without strangeness neutrality,respectively. μ B / T = μ B / T = μ B / T = μ B / T = μ B / T = μ B / T =
60 80 100 120 140 160 180 20002468 T [ MeV ] μ Q , S N [ M e V ] FIG. 6. Temperature dependence of the electric chemical po-tential along lines of constant µ B /T = 0 . −−