Strangeness neutral equation of state with a critical point

Jamie M. Stafford

In collaboration with: Debora Mroczek, Angel Nava, Jaki Noronha-Hostler, Paolo Parotto, Damien Price, Claudia Ratti 3

and δ = 5, which are within few percent of their exact T, GeV QGP values in three dimensions. The result of Eq. (9) can then critical be simplified to t point

2 4 6 8 81 783θ +105θ 5θ +2θ H κ4(t, H)= 12 − 14 3 2 3 − 2 5 . (10) − R / (3 θ ) (3 + 2θ ) 0.1 − freezeout hadron gas curve We represent κ4(t, H) graphically as a density plot in nuclear Fig. 1. We see that the 4-th cumulant (and kurtosis) matter is negative in the sector bounded by two curved rays 0 1 µ , GeV H/tβδ = const (corresponding to θ 0.32). B ± ≈ ± FIG. 2: A sketch of the phase diagram of QCD with the freeze- out curve and a possible mapping of the Ising coordinates t and H.

The negative minimum is small relative to the positive peak, but given the large size of the latter, Ref.[7, 15], the negative contribution to kurtosis may be significant. In addition, the mapping of the freezeout curve certainly Search for Criticality need not be H = const, and the relative size of the posi- tive and negative peaks depends sensitively on that. The trend described above appears to show in the re- ➤ Ongoing search for critical point requires support from theory community to provide cent lattice data, Ref.[10], obtained using Pade resum- mation of the truncated Taylor expansion in µB.Asthe candidates for criticality-carrying observables chemical potential is increased along the freezeout curve, the 4-th moment of the baryon number fluctuations be- gins to decrease, possibly turning negative, as the critical (a) point is approached (see Fig.2 in Ref.[10]). 120 Another observation, which we shall return to at the 100 end of the next section, is that κ4 grows as we approach 80 the crossover line, corresponding− to H =0,t>0onthe 60 4

Κ diagram in Fig. 1(a). On the QCD phase diagram the 40 20 freezeout point will move in this direction if one reduces 0 the size of the colliding nuclei or selects more peripheral ! collisions (the freezeout occurs earlier, i.e., at higher T , 20! ! 0.4 0.2 0.0 0.2 0.4 0.6 in a smaller system). t (b) EXPERIMENTAL OBSERVABLES FIG. 1: (color online) (a) – the density plot of the function NSAC 2015 Long Range Plan for Nuclear Physics κ4(t, H)givenbyEq.(10)obtainedusingEq.(9)forthelinear M. Stephanov, K. Rajagopal and E. Shuryak, PRD (1999) In this section we wish to connect the results for the parametric modelM. Eqs.Stephanov, (6), PRL (7), (2011) (8) and β =1/3, δ =5.The2 κ4 < 0regionisred,theκ4 > 0–isblue.(b)–thedependence fluctuations of the order parameter field σ to the fluctua- of κ4 on t along the vertical dashed green line on the density tions of the observable quantities. As an example we con- plot above. This line is the simplest example of a possible sider the fluctuations of the multiplicity of given charged mapping of the freezeout curve (see Fig. 2). The units of t, particles, such as pions or protons. H and κ4 are arbitrary. For completeness we shall briefly rederive the results of Ref.[7] using a simple model of fluctuations. The model Also in Fig. 1 we show the dependence of κ4 along a captures the most singular term in the contribution of the line which could be thought of as representing a possible critical point to the fluctuation observables. Consider a mapping of the freezeout trajectory (Fig. 2) onto the tH given species of particle interacting with fluctuating crit- plane. Although the absolute value of the peak in κ4 ical mode field σ. The infinitesimal change of the field δσ depends on the proximity of the freezeout curve to the leads to a change of the effective mass of the particle by critical point, the ratio of the maximum to minimum the amount δm = gδσ. This could be considered a def- along such an H = const curve is a universal number, inition of the coupling g. For example, the coupling of approximately equal to 28 from Eq. (10). protons in the sigma model is gσpp¯ . The fluctuations δfp − Equation of State with Criticality

➤ Update to the original EoS that first matched the Taylor expansion coefficients from Lattice QCD and implemented critical features based on universality arguments

PHYSICAL REVIEW C 101,034901(2020)

QCD equation of state matched to lattice data and exhibiting a critical point singularity

Paolo Parotto ,1,2,* Marcus Bluhm,3,4 Debora Mroczek,1 Marlene Nahrgang,4 J. Noronha-Hostler,5 Krishna Rajagopal,6 Claudia Ratti,1 Thomas Schäfer,7 and Mikhail Stephanov8 1Department of Physics, University of Houston, Houston, Texas 77204, USA 2Department of Physics, University of Wuppertal, Wuppertal D-42219, Germany 3Institute of Theoretical Physics, University of Wroclaw, 50204 Wroclaw, Poland 4SUBATECH UMR 6457 (IMT Atlantique, Université de Nantes, IN2P3/CNRS), 4 rue Alfred Kastler, 44307 Nantes, France 5Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 6Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 7Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA 8Physics Department, University of Illinois at Chicago, Chicago, Illinois 60607, USA

(Received 21 December 2018; revised manuscript received 26 November 2019; accepted 7 February 2020; published 2 March 2020)

We construct a family of equations of state for QCD in the temperature range 30 MeV ! T ! 800 MeV and in the chemical potential range 0 ! µB ! 450 MeV. These equations of state match available lattice QCD 4 results up to O(µB ) and in each of them we place a critical point in the three-dimensional (3D) Ising model universality class. The position of this critical point can be chosen in the range of chemical potentials covered by the second Beam Energy Scan at the Relativistic Heavy Ion Collider. We discuss possible choices for the free parameters, which arise from mapping the Ising model onto QCD. Our results for the pressure, entropy density, baryon density, energy density, and speed of sound can be used as inputs in the hydrodynamical simulations of the fireball created in heavy ion collisions. We also show our result for the second cumulant of the baryon number in thermal equilibrium, displaying its divergence at the critical point. In the future, comparisons between Code can be downloaded at: RHIC data and the output of the hydrodynamic simulations, including calculations of fluctuation observables, built upon the model equations of state that we have constructed may be used to locate the critical point in the https://bitbucket.org/bestcollaboration/ QCD phase diagram, if there is one to be found. eos_with_critical_point/src/master/ DOI: 10.1103/PhysRevC.101.034901

3 I. INTRODUCTION Lattice QCD simulations cannot currently be performed at finite density. For this reason, a first principle prediction of The search for a possible QCD critical point is receiving the existence and position of the critical point is still missing. increasing attention, which will culminate in the second Beam Several QCD-based models predict its location on the phase Energy Scan (BES-II) at the Relativistic Heavy Ion Collider diagram, which naturally depends on the model parameters (RHIC) at Brookhaven National Laboratory. The main goal of and approximations (for a review, see, e.g., Ref. [4]). This the BES-II program is to discover a critical point, or constrain aspect makes the critical point search challenging, and is at the its location, on the phase diagram of strongly interacting mat- basis of the systematic scan in collision energies performed ter. One of the central questions that these experiments aim at RHIC. We anticipate that nonmonotonous dependence of to answer is whether the continuous crossover [1]between specific observables on collision energy will indicate the pres- quark-gluon plasma and hadronic matter that occurs as a ence of the critical point as the freezeout point traverses the function of decreasing T at µ 0turnsintoafirst-order B critical region [5,6]. As the BES-II approaches, it is therefore phase transition above some critical= point at a nonzero µ , B important to predict the effects of the critical point on several corresponding to heavy ion collisions below some collision observables. energy [2,3]. One of the main theoretical approaches to pursue this goal is represented by hydrodynamical simulations of the evolution of the fireball produced in heavy ion collisions (see, e.g., *Corresponding author: [email protected] Ref. [7] and references therein). While modifications of the hydrodynamical approach itself are needed in the vicinity of Published by the American Physical Society under the terms of the the critical point [8–11], the equation of state (EoS) used as Creative Commons Attribution 4.0 International license. Further an input in these simulations needs to reflect all theoretical distribution of this work must maintain attribution to the author(s) knowledge currently available as well as contain the singular- and the published article’s title, journal citation, and DOI. Funded ity associated with the QCD critical point at a parametrically by SCOAP3. controllable location. Thus, the purpose of this paper is to

2469-9985/2020/101(3)/034901(18) 034901-1 Published by the American Physical Society Equation of State with Criticality

➤ This equation of state has been used to study off-equilibrium effects on initial conditions and the sign of the kurtosis in heavy-ion collisions

T. Dore et al, PRD (2020) D. Mroczek et al, PRC (2021) 4 HIC Phenomenology

➤ Modeling should mimic experimental conditions in all stages in order to provide robust comparisons and estimates

ICCING: M. Martinez et al, arXiv: 1911.12454 & 1911.10272 BQS diffusion: J. Fotakis et al, PRD (2020) MUSIC: G. Denicol et al, PRC (2018) BEShydro: L. Du and U. Heinz, Comp. Phys. Comm (2020) BQS fluctuations: V. Vovchenko et al, PLB (2021)

5 New (!) Equation of State with Criticality

➤ New version includes (but not limited to): ➤ Imposing conditions on conserved charges as present ⟨nS⟩ = 0

in HICs ⟨nQ⟩ = 0.4⟨nB⟩ 10 ➤ Hadronic species present in SMASH hadronic transport approach

FIG. 4. Isentropic trajectories in the (T, µB ) plane, for s/nB =420, 144, J.70 Noronha-Hostler,, 30, corresponding JS et to al, collision PRC (2019) ener- gies psNN =200, 62See.4, also:27, 14 A..5 Monnai GeV respectively. et al, PRC The(2019) solid black lines correspondH. toElfner,nS RHIC-BES=0, nQ =0 Seminar.4 nB (2020)while the dashed red lines to µ = µh =0.i h i h i J.S WeilQ et al, PRC (2016) 6 Taylor Coefficients from LQCD

➤ 4 Lattice results for Taylor expansion of pressure around μB = 0 up to #(μB) are the backbone of the procedure for creating this equation of state n P(T, μB) μB 2 = c (T) T4 ∑ 2n ( T ) n

1.6 3.5 HRG μS = μQ=0 1.4 HRG Str. Neutr. 3.0 1.2 Lat μS = μQ=0 2.5 Lat Str. Neutr. 1.0 3 0 0

1 2.0 0.8 1 2 4

c 1.5 0.6 c

0.4 1.0

0.2 0.5

0.0 0.0 0 100 200 300 400 500 0 100 200 300 400 500 T (MeV) T (MeV) - original BES-EoS J. Guenther et al, NPA (2017) updated version R. Bellweid et al, PRD (2015) - See also: A. Bazavov et al, PRD (2017) 7 Lattice EoS at Finite T & μB

➤ Strangeness neutral equation of state from first-principles can be produced by running in “LAT” mode

8 4. Theory and Phenomenology of the Critical Point

4.1. Critical phenomena The critical phenomena which we shall focus on occur at an end-point of a first-order transition in a thermo- dynamic system. The first-order transition corresponds to a situation when a thermodynamic system under given external conditions (such as T and µ, for example) can be in equilibrium in two distinct thermodynamic states. Such a two-phase coexistence can occur only for special values of external parameters, typically, on a manifold of one less dimension than the space of external parameters. E.g., in the T µ plane this manifold is a first-order transition line. One of the two states is thermodynamically stable on one side of the first-order phase transition, and the other – on the other side. By adjusting parameters along the phase-coexistence line one could arrive at a special point where the di↵erence between the two coexisting phases disappears. This is a critical point, also known as a second-order phase transition. This point is characterized by critical phenomena which manifest themselves in singular thermodynamic and hydrodynamic properties. The two most common examples of such critical points are the end point of the liquid-gas coexistence curve and the Curie point in a (uniaxial) ferromagnet. Although the two systems in which these two examples occur are di↵erent on a fundamental, microscopic level, the physics near the critical point is remarkably similar on qualitative as well as quantitative level. This observation is the basis of the concept of universality of the second-order phase transitions. The uniaxial, or Ising, ferromagnet is the simplest of such systems. It can be modeled by a lattice of spins s = 1, or two-state systems, with local (e.g., nearest neighbor) interaction favoring the alignment of spins in the i ± same direction. There are two ground states, with all the spins pointing in one of the two possible directions. The degeneracy is lifted if one applies a magnetic, or ordering, field h, which changes the energy of the spins by h i si. The two ordered states are distinguished by the value of the magnetization P 1 N M = s (18) N i Xi=1 which equals +1 or 1 depending on the sign of h, or more precisely, by its thermal average M . At finite, low h i enough temperature the ordering persists and M plays the role of the order parameter which flips sign at h = 0. The h i two ordered phases coexist on the line h = 0 in the T h plane as shown in Fig. 7 Universal Scaling EoS The magnetization M along the coexistence line, h = 0, decreases with increasing temperature due to thermal fluctuations. At the Curie temperature, Tc, the magnetization completely vanishes and remains zero for all higher ➤ temperatures. The coexistence line (the first-order phase transition) ends at T = Tc – the critical point. There is only Criticality is implemented by mappingone phasethe atcritical and above point the Curie from point temperature. the 3D Ising model onto the QCD phase diagram Similarly, liquids (e.g., water) coexists with their vapour at given pressure p at the boiling temperature T, which defines a line in the T vs p plane. At any of the coexistence points on this line the molecules making up the substance ➤ Relevant and analogous quantities for Ising-QCD map:

➤ Magnetic field, h Baryon chemical potential, µ

➤ Magnetization, M Baryon density, nB

T − TC ➤ Reduced temperature: t = TC

➤ Gibbs’ free energy/thermodynamic potential = −Pressure

Figure 7: The phase diagram of the Ising ferromagnet. The transition between the phases with positive and negative magnetization is a first-order transition for T < Tc and a continuous crossover at T > Tc. The transition changes its character at the critical point. K. Rajagopal and F. Wilczek, Nucl. Phys. B (1993) P. Parotto et al, PRC (2020)19 A. Bzdak et al, Phys. Rep. (2020) 9 C. Nonaka, M. Asakawa, PRC (2005) 2 by using the SMASH hadronic list as input in the HRG of µB spanning positive and negative values. Fur- model. Because SMASH has become a standard trans- thermore, the equations defined here are subject to port code used within the field, we ensure consistency the following constraints on the parameters: R across all stages of phenomenological modeling of heavy- 0, ✓ ✓0 1.154. ion collisions. We begin by describing the general proce- | |  ⇠ dure for developing an EoS with a critical point in the 3D 2. Choose the location of the critical point and map Ising model universality class. We then provide details the critical behavior onto the QCD phase diagram via a linear map from T , µ to r, h : of the implementation of the new features into the EoS. { B} { } In order to study theMapping e↵ect of a critical point thatthe could 3D Ising Model onto QCD T Tc potentially be observed during the BES-II at RHIC on = !(⇢r sin ↵1 + h sin ↵2) (3) QCD thermodynamics, we utilize the 3D Ising model to Tc map such critical behavior➤ Phase onto the transition phase diagram along of Ising temperature axis fixed onto QCD phase diagram along QCD. The 3D Ising model was chosen for this approach µB µB,c because it exhibits the same scaling features in the vicin- = !( ⇢r cos ↵1 h cos ↵2) (4) transition line from LQCD Tc PAOLO PAROTTO et al. ity of a critical point as QCD, in other words they belongPHYSICAL REVIEW C 101,034901(2020) to the same universality class [49, 50]. We implement the where (T ,µ ) are the coordinates of the critical non-universal mapping of the 3D Ising model onto the c B,c point, and (↵1,↵2) are the angles between the axes QCD phase diagram in such a way that the Taylor ex- of the QCD phase diagram and the Ising model pansion coecients of our final pressure match the ones ones. Finally, ! and ⇢ are scaling parameters for calculated on the lattice order by order. This prescrip- the Ising-to-QCD map: ! determines theT − overallT tion can be summarized as follows: scale of both r and h,while⇢ determines the rela-C = w (t ρ sinα1 + h sinα2) 1. Define a parametrization of the 3D Ising model near tive scale between them. TC the critical point, consistent with what has been t previously shown in the literature [6, 43, 51, 52], 3. As previously established in Ref. [43], we reduce the number of free parameters from sixμB to− four,μB,C by M = M0R ✓ assuming the critical point sits on the chiral phase= − w (t ρ cosα1 + h cosα2) transition line, and by imposing that the TrCaxis of h = h Rh˜(✓) (1) 0 the Ising model is tangent to the transition line of r = R(1 ✓2) QCD at the critical point: where the magnetization M, the magnetic field h, µ 2 and the reduced temperature r, are given in terms T = T +  T B + (µ4 ). (5) 0 0 T O B of the external parameters R and ✓. The normal- ✓ 0 ◆ ization constants for the magnetization and mag- In this study, we maintain consistency with the netic field are M =0.605 and h =0.364, re- 0 0 original EoS development of Ref. [43] by utiliz- FIG. 2.spectively, Nonuniversal map=0 from.326 Ising and variables =4 (r.,8h) are to QCD critical coordinates ex- (T,µB ). ing the same parameters1. The critical point lies ponents in the 3D Ising Model, and h˜(✓)=✓ (1- 2 4 at Tc 143.2 MeV, µB,c=350 MeV ,whilethe 10 0.76201✓ +0.00804✓ ). { ' } P. Parotto et al, PRC (2020) which can be visualized in Fig. 2.Thismapmakesuseof IV. THERMODYNAMICSangular parameters are orthogonal ↵1=3.85°and six parameters, two of which correspondThe singular to the part location of the of pressure is described by ↵ =93.85 , and the scaling parameters are !=1 and A. Strategy 2 ° the critical point on the QCD phasethe parametrized diagram, two Gibbs’ are the free energy: ⇢=2. However, we remind the reader that such a angles that the r and h axes form with the T const. lines, The strategy we wish to pursue in orderchoice to produce of parameters an equa- only has an illustrative pur- PIsing == G(R, ✓) and (w, ρ)arescalefactorsforthevariablesr and h. While tion of state for QCD(2) which meets thepose, requirements and that stated we do in not make any statement about 2 ↵ ˜ Sec. I is the following. Starting from the Taylor expansion w represents a global scaling for the Ising variables,= h0M0 namelyR (✓h(✓) g(✓)), the position of the critical point or the size of the coefficients up to O(µ4 )inEq.(1), available from lattice determining the size of the critical region, ρ represents a B critical region. As this framework does not serve to where QCD simulations, we rewrite them as a sum of an “Ising” relative scaling of r and h,thusroughlydeterminingtheshape yield a prediction for the critical point, but rather of it. 2 contribution2 2 coming2 3 from the critical point of QCD and a g(✓)=c0 + c1(1 ✓ )+c2(1 ✓ ) + c3(1 ✓ ) , to provide an estimate of the e↵ect of critical fea- At this point, we have a double map between coordinates: “non-Ising” contribution, which would contain the regular part as well as any other possible criticalitytures on presentheavy-ion-collision in the systems, the users can c0 = (1 + a + b), pick their preferred choice of the parameters and (R, θ ) (r, h) 2(T,µ↵ ), (13) region of interest: B test its e↵ect on observables. In particular, we note !−→ ←→ 1 1 4 LAT 4 non-Ising Ising c1 = ((1 2)(1 + a +T bc)n 2(T)(a +2T cbn)), (T ) f (Tthat,µB by0) varyingcn (T ), the(16) parameters ! and ⇢ it is possi- 2 ↵ 1 = + = where the second step is globally invertible. We will now ap- ble to increase or decrease the e↵ects of the critical ply the thermodynamics we developed in1 the previous section where f (T,µB )isaregularfunctionofthetemperatureand for the Ising model, making usec2 of= the additional(2b (1 variables2)(a +2chemicalb)), potential, with dimension of energy to the fourth 2↵ power. Away from the critical regime, f just reshuffles the (R, θ ), to the QCD phase diagram, in a parametrized1 form given by Eqs. (11)and(12). c3 = b(1 2). regular terms and can be chosen arbitrarily. Near the critical 2(↵ + 1) 1 In order to do this analytically, we would need the map point, the choice for f is almostAs in arbitrary, Ref. [43], with we assume the only the transition line to be a parabola, requirement being that it must notand add utilize any the leading curvature singular parameter  = 0.0149 from Ref. [4]. (R, θ ) (T,µ ), which unfortunately cannot be globally B Because QCD is symmetric about µB = 0, we re- Recent results from lattice QCD [33, 53] are consistent with this inverted.!−→ Therefore, it is necessary to solve the following behavior. In general, though, any term in f beyond a constant quire that PIsing is also matter-anti-matterintroduces sub-leading symmet- behaviorvalue, in the and vicinity predict of the the critical next to leading order parameter 4 to be relations numerically: consistent with 0 within errors. ric. Thus, we perform the calculationspoint. For this in reason, a range the simplest choice is to take f to be a constant, with the appropriate dimension. This also ensures T (R, θ ) Ti 0, (14) − = that no subleading behavior is introduced near the critical µB(R, θ ) µBi 0, (15) point, which cannot be predicted through universality. Note − = non-Ising that Eq. (16)istobeunderstoodasadefinitionforthecn for each value of (T,µB ) needed in the QCD phase diagram. coefficients. We proceed in the following way: We choose a range of Once these coefficients are obtained, we will build a interest for T and µB,andgivenachoiceoftheparametersin Taylor expansion in µB analogous to the lattice one, using the Ising-QCD map, we solve Eqs. (14)and(15)numerically the “non-Ising” coefficients. The latter have the advantage for a two-dimensional grid in T and µB in the desired range, that the critical behavior coming from the critical point has thus providing a discrete inverse map (T,µB ) (R, θ ). been removed, so that the expansion can be pushed to larger !−→ With this solution, although not analytic, it is possible to values of µB.Thisprovidesanexpressionforthe“non-Ising” transport the thermodynamics of the Ising model [written in pressure over a broad region of the QCD phase diagram. The terms of (R, θ )], into the QCD phase diagram, given a choice assumption here is that the Ising critical point contribution to of parameters for the map. the Taylor coefficients from lattice QCD can be reproduced

034901-4 3D Ising Model Parametrization

➤ Universal scaling behavior encoded in parameters (R, θ):

➤ βδ Magnetic field: h = h0R H(θ) ➤ Reduced temperature: t = R(1 − θ2)

➤ β Magnetization: M = M0R θ

➤ 2−α Gibbs’ free energy: G = h0M0R [g(θ) − θH(θ)]

where α = 0.11, β = 0.326, δ = 4.8 are 3D Ising critical exponents, H(θ) is a polynomial in odd powers of θ, and g(θ) is a polynomial in (1-θ2).

➤ Generally, free energy includes singular and non-singular contributions:

➤ P(T, μB) = − G[R, θ] + Pbkg(T, μB) P. Parotto et al, PRC (2020) A. Bzdak et al, Phys. Rep. (2020) C. Nonaka, M. Asakawa, PRC (2005) J. Zinn-Justin Quantum Field theory 11 and Critical Phenomena PAOLO PAROTTO et al. PHYSICAL REVIEW C 101,034901(2020)

Singular and Non-singular Contributions

FIG. 4. Parametrization of baryon susceptibilities from lattice QCD [20,64]andHRGmodelcalculations. ➤ Our construction requires that the total free energy (pressure) is the one from the lattice, so order-by-orderthe most we up-to-date have: particle list available from the Particle in unphysical values for the thermodynamic observables. The Data Group [66](listPDG2016 in Ref. [67]). recipe to cure this problem is to make use of the fact that one Lat Ising + Non−Ising The smoothχN (T curves) = χN obtained(T) + fromχN the parametrization(T) will can reasonably expect the system to find itself in a hadron gas LAT be the cn (T )coefficientsinEq.(16), thus defining the state in that region of the phase diagram and find a way to 0.25 0.05 4 Non-Ising LAT cn (T ) coefficients that will be used for the Taylor ex- smoothly mergeχN the pressure coming from the procedure we 0.20 0.04 χIsing 3 developed so farN with the one from the HRG model. pansion. Figure 5 shows the comparison of the “Ising” and T 4 0.15 0.03 c χNon − Ising 2 “non-Ising” contributions to the parametrized lattice/HRG The smoothT 4 mergingN can be obtained through a hyperbolic 0.10 0.02 χ0 χ2 χ4 1 tangent as model results.0.05 0.01 0 0.00 0.00 PFinal(T,µB ) P(T,µB ) 1 T T "(µB ) −1 −0.05 −0.01 1 tanh − VI. RESULTS 4 4 −2 T = T 2 + !T " −0.10 −0.02 $ % &' 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 T (MeV) At this point, we have all theT (MeV) ingredients in Eq. (17): PHRG(T,µB ) 1 T T (µB ) T (MeV) 1 tanh − " , + T 4 2 − !T 2 " 2n $ % &' 4 non-Ising µB 4 Ising P(T,µB ) T c (T ) T P (T,µB ), (33) = 2n T + C symm n 0 " # != (32) where T "(µB )worksasthe“switchingtemperature”and!T " JS et al, arXiv:2103.08146 12 which is now straightforward. However, although the overall is roughly the size of the “overlap region” where both pres- behavior is correct, at low temperatures and in particular sures contribute to the sum. The dependence on the baryon in regions where the ratio µB/T is very large, the pressure chemical potential of the “switching temperature” is chosen becomes negative, and so do other observables as well. This is to be parabolic and parallel to the chiral transition line we due to the fact that, given our choice of the function f (T,µB ) assumed in Eq. (26): in Eq. (17), the “Ising” coefficients at low temperature follow κ 2 apowerlaw,whereasthefullonesfromlatticecalculations T "(µB ) T0 µ T ∗, = + T B − fall off exponentially; hence, there will always be a value of T 0 non-Ising for which one or more of the cn (T )fallsbelowzero,and where T0 and κ are the transition temperature and curvature thus a value of µB/T large enough that the pressure from the of the transition line at µB 0, and we choose T ∗ 23 MeV Taylor expansion in Eq. (17)islargeandnegative,resulting and in Eq. (33) !T 17 MeV.= = " = 034901-8 EoS Thermodynamic Outputs

➤ Pressure and its derivatives show effects of critical region on these quantities: stronger effects with increasing derivatives

Pressure B χ2

Baryon density JS et al, arXiv:2103.08146 13 EoS Thermodynamic Outputs

➤ Energy density and entropy exhibit discontinuities, while the speed of sound approaches zero at the critical point

Energy density Speed of sound

Entropy JS et al, arXiv:2103.08146 14 Isentropic Trajectories

➤ Isentropes show the path of the HIC system through the phase diagram in the absence of dissipation

400 S /nB = 68 S /nB = 25

S /nB = 50 S /nB = 23 350 S /nB = 42 S /nB = 21 S /nB = 35 S /nB = 18

S /nB = 28 S /nB = 16 300

250

T (MeV) 200

150

100

50 0 100 200 300 400 500 μB (MeV)

JS et al, arXiv:2103.08146 15 Isentropic Trajectories

➤ Isentropes show the path of the HIC system through the phase diagram in the absence of dissipation

➤ Different path when conserved charge conditions applied

Str. Neutr. μS = μQ = 0 600 S /nB = 68 S /nB = 68 S /nB = 35 S /nB = 35

S /nB = 21 S /nB = 21 500

400

T (MeV) 300

200

100

100 150 200 250 300 350 400 450 500 μB (MeV) JS et al, arXiv:2103.08146 16 Correlation Length

➤ Additionally, calculate the correlation length in the 3D Ising model:

ν β ξ2(t, M) = f 2 |M|−2 / g(x)

where f =1fm, ν = 0.63 is the correlation length critical exponent, g(x) is the scaling function and the scaling |t| parameter is x = |M|1/β

Correlation length

B. Berdnikov and K. Rajagopal, PRD (2000) C. Nonaka and M. Asakawa, PRC (2005) JS et al, arXiv:2103.08146 17 Conclusions

➤ We provide an update to the BES-EoS that includes strangeness neutrality conditions, includes the same hadronic states as SMASH, and performs in a range of temperature and baryonic chemical potential relevant for BES-II.

Str. Neutr. μS = μQ = 0 600 S /nB = 68 S /nB = 68 S /nB = 35 S /nB = 35

S /nB = 21 S /nB = 21 500 ➤ We see the expected critical features in the EoS 400

and note a shift in the isentropic trajectories T (MeV) 300 between the new and original versions. 200 100

100 150 200 250 300 350 400 450 500 μB (MeV) ➤ A calculation of the correlation length in the 3D Ising model is provided.

18