(2021) Chiral Phase Transition in an Expanding Quark System

PHYSICAL REVIEW C 103, 014901 (2021)

Chiral phase transition in an expanding quark system

Ziyue Wang ,1,* Shuzhe Shi ,2 and Pengfei Zhuang1 1Physics Department, Tsinghua University, Beijing 100084, China 2Department of Physics, McGill University, 3600 University Street, Montreal, Quebec H3A 2T8, Canada

(Received 6 August 2020; accepted 18 December 2020; published 8 January 2021)

We investigate the influence of chiral symmetry which varies along the space-time evolution of a system, by considering the chiral phase transition in a nonequilibrium expanding quark-antiquark system. The chiral symmetry is described by the mean field order parameter, whose value is the solution of a self-consistent equation, and affects the space-time evolution of the system through the force term in the Vlasov equation. The Vlasov equation and the gap equation are solved concurrently and continuously for a longitudinal boost-invariant and transversely rotation-invariant system. This numerical framework enables us to carefully investigate how the phase transition and collision affect the evolution of the system. It is observed that the chiral phase transition gives rise to a kink in the flow velocity, which is caused by the force term in the Vlasov equation. The kink is enhanced by larger susceptibility and tends to be smoothed out by nonequilibrium effects. The spatial phase boundary appears as a “wall” for the quarks, as the quarks with low momentum are bounced back, while those with high momentum go through the wall but are slowed down.

DOI: 10.1103/PhysRevC.103.014901

I. INTRODUCTION it possible to search for the CEP in the QCD phase diagram [9–11]. Besides the ongoing BES program at the BNL Rela- One of the major motivations in the study of finite- tivistic Heavy Ion Collider (RHIC), several other programs at temperature quantum chromodynamics (QCD) is to shed light other facilities such as the GSI Facility for Antiproton and Ion on the phase structure of the strong interaction in matter. Research (FAIR) and the Nuclotron-based Ion Collider Facil- Lattice-QCD data have confirmed that the chiral and decon- ity (NICA) have also contributed to the search for the QCD finement phase transition is a crossover for small baryon critical end point. The related experiments are mainly driven chemical potential [1,2]. The sign problem at large baryon by measurements of net-proton or net-charge multiplicity fluc- chemical potential [3,4] prevents lattice QCD from giving tuations [12–14] which are expected to show characteristic precise predictions about the phase transition at finite density. nonmonotonic behavior near the phase transition and espe- Model calculations based on the Nambu–Jona-Lasinio (NJL) cially near the CEP [15,16]. The fast dynamics in the fireball model, the quark meson model, and various beyond mean field renders it difficult to bridge the gap between the experiment frameworks have all predicted that the chiral phase transition and the theories since, from the theoretical aspect, the QCD at finite density is a first-order phase transition [5–8]. Ther- phase structure is investigated in an equilibrium, long-lived, modynamic theory then predicts a critical end point (CEP) extremely large, and homogeneous system. In contrast, the between the crossover and the first-order phase transition, fireball in the heavy ion collision is a highly dynamical sys- which is a second-order phase transition. However, due to tem, characterized by very short lifetime, extremely small various approximations adopted in the model calculations, size, and fast dynamical expansion. The finite-size effect and there is not an agreement on the location of the CEP on the off-equilibrium effect prevent a divergent correlation length, phase diagram. and thus weaken the critical phenomenon which takes place The exploration of the QCD phase diagram is also one in the equilibrium system. On one hand, the fast expansion of the most important goals for relativistic heavy-ion exper- and cooling during the evolution of the fireball tends to drive iments. Through a systematic measurement over a range of the system out of local equilibrium. On the other hand, the beam energies, the beam energy scan (BES) program makes relaxation time diverges around the critical point [17], and the critical slowing down renders it harder for the system to reach local equilibrium. A thorough understanding of phase tran- *[email protected] sitions in the dynamical environment is thus of fundamental necessity to make profound predictions from the BES project. Published by the American Physical Society under the terms of the Various models have been applied to study the chiral Creative Commons Attribution 4.0 International license. Further phase transition and related critical phenomenon in an out-of- distribution of this work must maintain attribution to the author(s) equilibrium system. In Refs. [18,19], the authors investigate and the published article’s title, journal citation, and DOI. Funded the chiral phase transition in a free-streaming quark-antiquark by SCOAP3. system by solving the Vlasov equation through the test

2469-9985/2021/103(1)/014901(12) 014901-1 Published by the American Physical Society ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021) particle method. The chiral fields are included consider- (NJL) Lagrangian model [25–27], ing their mean field values and their equations of motion. L = ψ¯ γ μ∂ − ψ + ψψ¯ 2 + ψ¯ γ τψ 2 , The Vlasov equation for the quark-antiquark system is also (i μ m0 ) G[( ) ( i 5 ) ] (2) analytically solved in Ref. [20], assuming constant quark where ψ = (u, d)T is the two-component quark field in the mass, and a shell-like structure at late evolution times in flavor space, m0 is the degenerate current mass of the quarks, the center-of-mass (CM) frame is discovered. In Ref. [21] and τ is the Pauli matrix in the isospin space. The transport the nonequilibrium and collision effects on the deconfine- equations can be derived from a first-principle theory or an ment phase transition are investigated by solving the Vlasov effective model in the framework of the Wigner function equation assuming Bjorken symmetry. The elastic two-body [28–33]. At the classical level, the quarks are treated as quasi- collisions for the quarks and antiquarks is included by simu- particles, and the chiral field is approximated by a mean field, lating a Vlasov-type equation with a Monte Carlo test particle hence the Vlasov equation is coupled to the gap equation approach [22–24]. Among the aforementioned works, al- [33]. The disoriented chiral condensate (DCC) [34–36]is though the force term is also considered in the test particle negligible here because it appears as a quantum effect, and method [18,19,22–24], its effect has not been carefully exam- the σ condensate appears as the mean field and couples to the ined, and we find it plays an important role in the evolution of Vlasov equation through the inhomogeneous quark mass. In the system around the phase transition. order to investigate the collisions and nonequilibrium effect, In order to study the chiral phase transition in the nonequi- we take the relaxation time approximation for the collision librium state, we investigate an expanding quark-anitquark terms. The distribution function of the quark/antiquark num- gas system by solving the coupled Vlasov equation as well as ber density f ±(t, x, p) satisfies the coupled Vlasov equation the gap equation. This paper is arranged as follows. In Sec. III, and gap equation, we introduce and analyze the coupled Vlasov equation and ∇ 2 gap equation which we are going to solve, and give the related ± rm ± p ± ∂t f ∓ · ∇p f ± · ∇r f = C[ f ], (3) thermal quantities that can be obtained as momentum integrals 2Ep Ep of distribution function. In Sec. III, we consider a longitudinal + − d3 p f (x, p) − f (x, p) boost-invariant and transverse rotational-symmetric system, m 1 + 2G = m , (4) π 3 0 and derive the transport equation under such condition. In (2 ) Ep Sec. IV, we present our numerical process to solve the coupled where the + (−) sign stands for quark (antiquark). In this equations and then analyze the numerical result. In Sec. V,we work, we take the relaxation time approximation for the colli- summarize this work and give a brief outlook. C =− ± − ± /τ ± sion term, [ f ] ( f feq ) θ , with feq representing the corresponding local-equilibrium distribution function, while τθ is the relaxation time. It is worth noticing that one can II. VLASOV EQUATION AND THERMAL QUANTITIES make the substitution f −(t, x, p) ≡ 1 − f −(t, x, −p), which + The partons in an off-equilibrium system with background follows the same equation of motion as f . The transport field and collisions can generally be described by the Vlasov equations can be further simplified by adopting the recom- + − + − equation, bination f = f + f and g = f − f . In such a way, the evolution of f and g can be separated. Since f − and f + sat- ± ± ± isfy the same transport equation, f (t, x, p) and g(t, x, p)also ∂t f ∓ F · ∇p f ± v · ∇x f = C[ f ]. (1) satisfy the same transport equation, while the gap equation depends only on f but not g. One thus solves the coupled The above equation is applicable when the external fields and transport equation of distribution functions f (t, x, p) and the interactions between the (quasi)particles are sufficiently g(t, x, p) as well as the gap equation for a finite density sys- weak, so each particle can be considered to be moving along tem, and solves the transport equation of distribution function a classical trajectory, punctuated by rare collisions. As an f (t, x, p) together with gap equation for a system with vanish example of an evolving global symmetry in an expanding baryon density: parton system, we here consider the chiral symmetry in an ∇ 2 f − f off-equilibrium quark-antiquark system. At the classical level, ∂ − rm · ∇ + p · ∇ =− eq , v = / t f p f r f the velocity is p Ep; the energy of the quasi-particle 2Ep Ep τθ = 2 + 2 is Ep p m(x) , where m is the effective mass of the (5) quark and antiquark, which is space-time dependent and is ∇ 2 − determined by the evolving chiral symmetry. The chiral mean rm p g geq ∂t g − · ∇pg + · ∇rg =− , field acts as a background field, and affects the motion of 2Ep Ep τθ = ∇ quarks through the gradient of the field energy F xEp, (6) which is a continuous force on the quarks. The mass of the d3 p f (x, p) d3 p 1 quasiparticles m(x) is no longer a free parameter but is de- m 1+2G − 2G = m . 3 3 0 termined by the space-time dependent chiral symmetry. The (2π ) Ep (2π ) Ep constituent quark mass serves as the order parameter of chiral (7) symmetry. Its temperature dependence in equilibrium can be found in lattice-QCD simulation [1,2] and other model cal- In the numerical procedures, we consider a zero density culations. We here consider the SU(2) Nambu–Jona-Lasinio system in this paper and solve both the transport equation

014901-2 CHIRAL PHASE TRANSITION IN AN EXPANDING QUARK … PHYSICAL REVIEW C 103, 014901 (2021) of f (t, x, p) and the gap equation, and eventually get the phase transition can be defined at the maximum susceptibility time and space dependence of the distribution function dm/dT; for a second-order phase transition, the susceptibil- and the quark mass. The second integral in the gap equation is ity diverges at critical point. In an equilibrium system, the β the vacuum part, which has ultraviolet divergence and needs order parameter displays critical scaling m ∝ [(Tc − T )/Tc] to be regularized. Here we take the hard cutoff regularization in the vicinity of a second-order phase transition which can only for the vacuum part, be described by critical exponent β. The divergence of the 3 susceptibility is expected to affect the transport phenomenon d p 1 pT dpT dpz 1 ∇ 2 · ∇ = 2π of the system through the force term rm p f , in which π 3 π 3 2 (2 ) Ep 0 0 (2 ) Ep the force is provided by the ingredient of mass ∇rm .In 1 an equilibrium system with zero baryon density, the mass is = ( m2 + 2 2 − m2 + 2 ) 2 determined by temperature alone, and the force can be ex- 4π 2 pressed as ∇rm = 2m(dm/dT)∇rT . Since the susceptibility m + 2 √ exhibits a peak around the phase transition, the temperature m ln ∇ + m2 + 2 gradient rT in a realistic system is nonzero; the force term √ is expected be very large at the phase transition point in the + m2 + 2 2 + (m2 + 2 )ln √ . (8) space-time. However, when the system is close to the second- m2 + 2 order phase transition, the relaxation time may diverge [17] and the critical slowing down takes place, making it harder The same cutoff = 496 MeV is adopted for the longitudinal for the system to reach local equilibrium. When the system momentum and the transverse momentum in the integral. The has not yet reached local equilibrium, the temperature is not NJL coupling constant is set to be G = 1.688/ 2 in order to well defined, and neither is the expression 2m(dm/dT)∇ T guarantee the quark mass m ≈ 300 MeV in the vacuum. The r of the force term. If the phase transition takes place in an momentum integral of the finite temperature part is free from out-of-equilibrium system, the aforementioned effects of the divergence, and is left unregularized. Under such choice of force term may have been overestimated. In the following, we parameters, the temperature dependence of the quark mass in study the chiral phase transition in both local equilibrium and an equilibrium system can be directly calculated from the gap out-of-equilibrium systems, controlled by the relaxation time. equation (7) by taking the Fermi-Dirac distribution. The quark Then we analyze the influence of the phase transition and the mass for the chiral limit m = 0 and the real case m = 3.7 0 0 force term on the evolution of the system. MeV are presented in Fig. 1; the critical temperature is about The thermodynamic quantities can be constructed from the 156 MeV at vanishing baryon chemical potential. distribution function. For single-component medium, positive It is necessary to point out that the quark motion in phase particle f + for example, the current and energy-momentum space is controlled by the general Vlasov equation; there is stress tensor are defined by no divergence in its solution, and therefore we do not use any 3 cutoff in the treatment of the Vlasov equation itself. The cutoff μ d p μ + J+ (t, x) = p f (t, x, p), is used only in the NJL model, which is employed to describe (2π )3E p the chiral symmetry, namely the mean-field force term in the 3 (9) μν d p transport equation. The cutoff = 496 MeV here is for the T (t, x) = pμ pν f +(t, x, p), + π 3 longitudinal and transverse momenta; its translation to the (2 ) Ep √ √ μ three-momentum cutoff is 4962 + 4962 = 2 × 496 MeV which should satisfy that the conservation laws ∂μJ = 0 μν ≈700 MeV which is the usually used cutoff in the standard and ∂ν T = 0. Since the total energy density and entropy NJL model. is the sum of that of positive particles and negative parti- The chiral phase transition at vanishing baryon chemical cles, while the net-quark number density is the difference of potential is a crossover in the real case, and is a second- positive particles and negative particles, the above equations order phase transition in the chiral limit. For a crossover, the can be rewritten using the redefined distribution functions f = f + + f − and g = f + − f −, 3 300 d p 0 Jμ(t, x) = pμg(t, x, p), (2π )3E 250 −5 p

) 3 200 μν d p μ ν −10 T (t, x) = p p f (t, x, p), dT / π 3E MeV 150 (2 ) p ψ ( −15

ψ 3

100 dm μ d p μ + + m S (t, x) =− p ( f (t, x, p)ln f (t, x, p) −20 3 50 (2π ) Ep + + 0 −25 + [1 − f (t, x, p)] ln[1 − f (t, x, p)] 0 50 100 150 200 250 0 50 100 150 200 250 − − − T (MeV) T (MeV) + f (t, x, p)ln f (t, x, p) + [1 − f (t, x, p)] × ln[1 − f −(t, x, p)]). (10) FIG. 1. Temperature dependence of the quark mass in the real case and the chiral limit for various baryon chemical potentials in Using the Landau frame deﬁnition, the ﬂuid velocity can μ the equilibrium state. be determined as the timelike eigenvector (u uμ > 0) of the

014901-3 ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021)

μ ν μ stress tensor T ν u = u , with energy density being the a zero chemical potential limit considered in this paper, one corresponding eigenvalue. One could further obtain the par- would automatically find g = 0, hence J = 0, n = neq = 0, μ ticle number density and entropy density as n = uμJ and μeq = 0. μ s = S uμ. The space-time evolution of (t, x), n(t, x), and uz(t, x) would give us a quantitative idea about how the system evolves. III. SYMMETRY AND SIMPLIFICATION By taking the relaxation time approximation for the colli- For numerical simplicity, we will focus on the longitudinal sion kernel, we also need the corresponding local-equilibrium boost-invariant and transversal rotational-symmetric systems, distribution function for any given time and space point, which is a good approximation for ultracentral relativistic heavy-ion collisions. Under such symmetries, two constraints ± , = 1 , feq (x p) ± μ+μ / (11) are applied to the system, and the distribution function f has e( uμ p eq ) Teq + 1 five degrees of freedom. We introduce a new set of coordinates In the above distribution, the temperature and chemical po- (τ,η,ρ,φ, p⊥,ξ,θ); the original coordinates and the new tential are determined by matching the energy and number ones can be transformed through the following relations: density, i.e., = eq and n = neq, where (u · p)2d3 p t = τ cosh η, p = m(τ,ρ)2 + p2 cosh(ξ + η), ≡ f (t, x, p), (12) t ⊥ eq 3 eq (2π ) Ep z = τ sinh η, p = m(τ,ρ)2 + p2 sinh(ξ + η), (u · p)d3 p z ⊥ n ≡ g (t, x, p), (13) eq 3 eq (2π ) Ep x = ρ cos φ, px = p⊥ cos(φ + θ ), 2 2 y = ρ sin φ, p = p⊥ sin(φ + θ ), (15) with Ep = m(Teq,μeq ) + p . Similarly, we define the equi- y librium limit of the entropy density as where ρ ∈ [0, +∞), p⊥ ∈ [0, +∞), and ξ ∈ (−∞, +∞). (u · p)d3 p s =− ( f + ln f + + (1 − f +)ln(1− f +) Under the new set of coordinates, the longitudinal boost eq 3 eq eq eq eq (2π ) Ep invariance and transversal rotational symmetry of the distri- − − − − bution function can be translated into being independence + f ln f + (1 − f )ln(1− f )). (14) eq eq eq eq of φ and η, namely ∂φ f = ∂η f = 0. The phase space of the The difference between seq and the actual entropy density s distribution function becomes (τ,ρ, p⊥,ξ,θ). The transport quantifies how close the system is to the equilibrium state. In equation (6) is then reduced to

p⊥ cos θ m(∂ρ m) cos θ 1 m(∂τ m) sin θ p⊥ m(∂ρ m) f − feq ∂τ f + ∂ρ f − ∂ f − tanh ξ + ∂ξ f − − ∂θ f =− , (16) p⊥ 2 2 Ep Ep τ p⊥ + m Ep ρ p⊥ τθ where feq is the equilibrium distribution. To clearly analyze the θ dependence of the distribution function, and to simplify the calculation, we take the Fourier expansion of f (τ,ρ, p⊥,ξ,θ) with respect to θ, and also the equilibrium distribution function feq(τ,ρ, p⊥,ξ,θ), ∞ f (τ,ρ, p⊥,ξ,θ) = a0(τ,ρ, p⊥,ξ) + 2 [an(τ,ρ, p⊥,ξ) cos(nθ ) + bn(τ,ρ, p⊥,ξ)sin(nθ )], n=1 (17) ∞ feq(τ,ρ, p⊥,ξ,θ) = A0(τ,ρ, p⊥,ξ) + 2 [An(τ,ρ, p⊥,ξ) cos(nθ ) + Bn(τ,ρ, p⊥,ξ)sin(nθ )], n= 1 τ,ρ, ,ξ ≡ π −1 π τ,ρ, ,ξ,θ θ θ where the Fourier coefficients are obtained by the definition an( p⊥ ) (2 ) −π f ( p⊥ ) cos(n )d and −1 π bn(τ,ρ, p⊥,ξ) ≡ (2π ) −π f (τ,ρ, p⊥,ξ,θ)sin(nθ )dθ, and similarly for An and Bn. In a realistic system, one can further expect its symmetry under reflection along the xˆ, yˆ,orzˆ direction. Under such a condition, one can show that the θ-odd components of f and feq vanish, bn ≡ Bn ≡ 0, as well as an(−ξ ) = an(ξ ), An(−ξ ) = An(ξ ). Substituting the Fourier expansion (17) back into the Vlasov equation (16), we then reduce the Vlasov equation to the transport equations of the corresponding Fourier components a , n 1 m(∂τ m) p⊥ m(∂ρ m) 1 p⊥ m(∂ρ m) a0 − A0 (∂τ a ) − tanh ξ + (∂ξ a ) + (∂ρ a ) − (∂ a ) + − a =− , 0 2 2 0 1 p⊥ 1 1 τ p + m Ep Ep Ep ρ p⊥ τθ ⊥ 1 m(∂τ m) p⊥ m(∂ρ m) (∂τ a ) − tanh ξ + (∂ξ a ) + ∂ρ (a − + a + ) − ∂ (a − + a + ) n 2 2 n n 1 n 1 p⊥ n 1 n 1 τ p + m 2Ep 2Ep ⊥ 1 p⊥ m(∂ρ m) an − An − − ((n − 1)an−1 − (n + 1)an+1 ) =− , (18) 2Ep ρ p⊥ τθ

014901-4 CHIRAL PHASE TRANSITION IN AN EXPANDING QUARK … PHYSICAL REVIEW C 103, 014901 (2021)

τ,ρ = where√ the energy of a quasiparticle is Ep( ) with the nonvanishing components 2 2 m (τ,ρ) + p⊥ cosh(ξ ), with mass m(τ,ρ) obtained ττ τ,ρ = τ,ρ, ,ξ τ,ρ 2 + 2 2 ξ , from the gap equation, which in the new coordinates T ( ) a0( p⊥ )[m( ) p⊥] cosh ( ) p becomes p⊥dp⊥dξ ηη 1 2 2 2 T (τ,ρ) = a0(τ,ρ, p⊥,ξ)[m(τ,ρ) + p⊥]sinh (ξ ), m 1 + 2Nd G [a0(τ,ρ, p⊥,ξ) − 1 = m0. τ 2 2(2π )2 p (19) τρ 2 2 T (τ,ρ) = a1(τ,ρ, p⊥,ξ)p⊥ m(τ,ρ) + p⊥ cosh(ξ ), The momentum integral in the gap equation only relates p to the zeroth component a , while the ﬁrst- and second-order 0 ρρ τ,ρ = τ,ρ, ,ξ + τ,ρ, ,ξ 2 / , Fourier components contribute to the currents and energy- T ( ) [a0( p⊥ ) a2( p⊥ )]p⊥ 2 p momentum tensor. φφ 1 2 The energy-momentum tensor is T (τ,ρ) = [a (τ,ρ, p⊥,ξ) − a (τ,ρ, p⊥,ξ)]p /2, 2 0 2 ⊥ ρ p ⎛ ⎞ (21) T ττ 0 −T τρ 0 ⎜ ⎟ ξ ⎜ 0 −τ 2T ηη 00⎟ where is the abbreviation for p⊥dp⊥d . For this T μ = ⎜ ⎟, (20) p 4(2π )2 ν ⎝T ρτ 0 −T ρρ 0 ⎠ energy-momentum tensor, one can explicitly write 00 0−ρ2T φφ down the ﬂow velocity and energy density, which are the timeline eigenvector and the corresponding eigenvalue:

= (T ττ − T ρρ + (T ττ + T ρρ )2 − 4(T τρ)2 )/2, / / T ττ + T ρρ 1 1 2 T ττ + T ρρ 1 1 2 uμ ≡{uτ , 0, uρ , 0}= + , 0, − , 0 , (22) 2 (T ττ + T ρρ )2 − 4(T τρ)2 2 2 (T ττ + T ρρ )2 − 4(T τρ)2 2 Noting that the velocity has only two nonzero components, uτ and uρ , the equilibrium distribution function (11) can be expressed as 1 feq(τ,ρ, p⊥,ξ,θ) = . (23) −1 2 + 2 ξ − −1 θ + exp Teq uτ m p⊥ cosh Teq uρ p⊥ cos 1

IV. NUMERICAL PROCEDURE AND RESULT our concern in this paper, since the temperature is not well defined in such an initial stage; the discussion of phase In the numerical procedures, we solve the finite difference transition is also questionable. We here discuss the evo- versions of transport equations. The distribution function is lution of the system from local equilibrium towards an discretized on a fixed grid in the calculation frame. The phase off-equilibrium state after the formation of quark gluon space is discretized as follows: take 200 points for ρ within plasma. A local equilibrium initial state is adopted, and a the range ρ/ρ ∈ [−3, 3], 100 points for p within p /T ∈ 0 T T 0 local temperature could be assigned. The large gradient in [0, 8], 100 points for ξ within the range ξ ∈ [0, 6]. The Fourier the initial condition drives towards the off-equilibrium state, expansion of distribution function f with respect to θ is taken while the collisions drive the system towards local equilib- with maximum n = 7 to guarantee convergence. To avoid the rium. In order to describe the hot chiral restored medium in numerical instability around τ = 0, we take the initial time the inner part and the cold chiral symmetry broken medium as τ = 0.5 fm; the time step in the evolution is taken to be 0 in the outer part, we choose a Gaussian temperature pro- dτ = 0.0005 fm to guarantee stability. The calculation at the file T (ρ) = T exp (−ρ2/ρ2 ) for the initial state, with T = discrete time step n + 1 involves only quantities at the previ- 0 0 0 300 MeV and ρ = 2 fm. For the local equilibrium initial ous time step. At each time step, we solve both the transport 0 state, the distribution function is the Fermi-Dirac distribution, equation and the gap equation, and eventually get the time and / ρ −1 τ ,ρ, ,ξ = Ep T ( ) + = space dependence of the distribution function and the quark a√0( 0 p⊥ ) 2(e 1) , where energy is Ep 2 + 2 ξ ρ mass. This numerical framework is verified to be reliable by m p⊥ cosh , and T ( ) is the above initial temperature comparing the result with the analytical solution in a spherical profile. One can easily check that all other Fourier compo- τ = symmetric system [20], as well as checking the conservation nents vanish, ai( 0 ) 0fori 1. In the real case the current = . of the particle number and energy-momentum. mass is chosen to be m0 3 7 MeV, and in the chiral limit we = The initial state of the fireball is a highly off-equilibrium have the current mass m0 0. system; the evolution towards local equilibrium quark gluon The chiral phase transition is characterized by the chi- plasma is also an interesting problem. However, this is not ral order parameter σ , or the quark constituent mass. The

014901-5 ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021)

FIG. 2. The energy density (top), entropy density (middle), and equilibrium temperature (bottom) in the expanding system. The left and right panels correspond to small and large relaxation times. The lines are rainbow colored, representing different evolution times, and the evolution time τ is in units of fm/c. In the figure of entropy density, the solid and dashed lines correspond to realistic and equilibrium entropies. constituent mass is generated by the gap equation at each entropy, and temperature are presented in Fig. 2. The left panel space-time point, and enters√ the transport equation through corresponds to the evolution with small relaxation time, and 2 2 three ways: the energy Ep = m + p⊥ cosh ξ, the evolution the right panel corresponds to those of large relaxation time. rate of constituent mass ∂τ m, and the spatial gradient of mass The lines are rainbow colored, which represents different evo- ∂ρ m. In order to illustrate the influence of the phase transition lution time: from the red line to the purple line we represent and the force term in the transport equation, we consider the initial distribution to the distribution at later time. the following three different conditions. First, when solving The initial condition of the system is chosen to be an equi- the transport equation and the gap equation concurrently, the librium distribution, with a Gaussian distribution temperature effects of the phase transition and the force terms are both profile; the inner part (small ρ) has higher temperature and the taken into consideration. Second, for a comparison, we solve outer part (large ρ) has lower temperature. With the expansion the transport equation alone and keep the quark mass as a of the system, the temperature of the core area gradually constant, for instance m = 150 MeV. In this case, there is decrease, and the temperature of the outer area increases. The no phase transition or force term. Third, in order to further initial large gradient of the energy density drives the system illustrate the influence of the force term, we solve the trans- away from the local equilibrium, while the collisions bring port equation and the gap equation at each step, but ignore the system back to equilibrium. The entropy tells whether the the force term in the transport equation, namely assuming system has reached local equilibrium, the solid line represent ∂τ m = 0 and ∂ρ m = 0. We also study the influence of the the realistic entropy density and the dashed lines represent out-of-equilibrium effect by comparing the results of different the entropy of the equilibrium state. Since the equilibrium relaxation times; for small relaxation time the system stays state takes the maximum entropy. When the collisions are not close to local equilibrium while for large relaxation time the strong enough, the system takes longer time in the out-of- system is away from equilibrium. equilibrium state, where the realistic entropy is smaller than that of the equilibrium state. If the relaxation time is small, A. Thermodynamical quantities the collisions are strong enough to keep the system at local First, we self-consistently solve the coupled transport equa- equilibrium, the entropy density of the state is the same as tion as well as the gap equation with finite current mass, and that of the equilibrium state. adopt different relaxation times. At each time step, the distri- For small relaxation time the system experiences a quasi- bution function is obtained by evolving the transport equation; thermal expansion with a small anisotropy, while for large the energy density is calculated by definition Eq. (21). The relaxation time the expansion drives the system out of equilib- temperature is obtained by matching the realistic energy den- rium with a huge anisotropy. This phenomenon can be clearly sity to that of the equilibrium state. The energy density, depicted from the difference in longitudinal and transverse

014901-6 CHIRAL PHASE TRANSITION IN AN EXPANDING QUARK … PHYSICAL REVIEW C 103, 014901 (2021)

zero. For the real case, we here take phase boundary as the space-time point where the equilibrium temperature is around the critical temperature and dm/dx takes the maximum. The phase transition points are marked by the dotted lines in the Fig. 4. The phase diagram of the equilibrium strong interaction in matter in the T -μ plane is a map of the chiral symmetry. For nonequilibrium quark matter, the chiral symmetry breaking and restoration can be described by a phase diagram in space-time, with the phase boundary hypersurface indicating where the symmetry changes in the space-time. In the chiral limit, the phase transition hypersurface can still be defined by μ x (mψ = 0) no matter whether the system is in local equilibrium or not. Figure 5 presents the hypersurface obtained by self-consistent solution of a chiral limit system, with different colored lines representing different relaxation times. The collisions cast the kinetic energy into internal energy, thus decelerating the expansion of the system. In the initial P /P = τ 2T ηη/T ρρ FIG. 3. Time evolution of the anisotropy L T at state (τ − τ0 )/τ0 = 0, the core area ρ/ρ0 < 0.8 is in the chiral ρ/ρ = 0 0 for different relaxation times. symmetry restored phase. With the expansion of the system, the core area gradually cools, and the area of chiral symmetry restored phase shrinks. The free streaming system expands the pressure. The time evolution of the pressure ratio PL/PT = 2 ηη ρρ fastest; after (τ − τ )/τ > 2.2, the chiral restored phase dis- τ T /T at ρ/ρ0 = 0 for different relaxation time is pre- 0 0 sented in Fig. 3. For small relaxation time, the system remains appears. While the system with small relaxation time expands slower, it takes a longer time for the chiral restored phase to isotopic in the expansion with PL/PT = 1. For large relaxation time, the ratio drops continuously with the expansion. Finally, disappear. It appears from the evolution of mass (Fig. 4) and at large enough evolution time the system will gradually return the phase boundary (Fig. 5) that the various relaxation times to the isotropic expansion. do not have an obvious influence on the quark mass. Since the order parameter describes the long-range correlation and the overall property of the system, while the collision is a B. Constituent mass and phase boundary local process in the system and is a short-range correlation, In the equilibrium state, the quark constituent mass mψ the collision does not have big impact on the global symmetry serves as the order parameter of the chiral symmetry, which of the system. tells us to what extent the chiral symmetry is broken. Here in an expanding quark-antiquark system, the space-time de- C. Kink in velocity pendent quark mass mψ (τ,ρ) signals the chiral symmetry in the space-time. In the initial state, the inner part of the system When solving together the transport equation and the gap has higher temperature and is in the chiral symmetry restored equation, kinks in the velocities uτ and uρ are discovered. phase, and the constituent mass is small; the outer part of the The velocity along the ρ direction vρ ≡ uρ /uτ for different system has lower temperature and is in the chiral symmetry current mass and relaxation time are also presented in Fig. 4. broken phase with large constituent mass. By solving together As we have mentioned above, the quark mass enters the the coupled transport equation and gap equation, the evolution transport equation through both the energy Ep and the force of constituent quark mass in the space-time can be obtained term ∇Ep · ∇p f . The gradient of the field energy acts as a self-consistently. The evolutions of both the real case and continuous force on the quarks, describing the interaction be- the chiral limit are investigated for large and small relaxation tween the quarks and the mean fields. This interaction changes times; the results are presented in Fig. 4. the quarks’ momenta. Although this force term is also consid- The upper two figures present the quark mass mψ (τ,ρ) ered in simulations such as the test-particle method [18,19,22– ρ τ and transverse velocity vρ (τ,ρ) ≡ (u /u ) in the real case, 24], its influence on the phase transition and the expansion has and the lower two figures correspond to those of the chiral not been carefully investigated. For a chiral phase transition at limit. With the expansion of the system, the quark mass in the low density, it is either a crossover or a second- order phase inner area grows with time, indicating the gradually restoring transition; in both cases the order parameter changes contin- chiral symmetry; the quark mass of the outer area decreases, uously. In comparison, the gradient of the order parameter indicating the breaking of chiral symmetry. In the equilibrium diverges at a second-order phase transition, hence the force chiral phase transition, the phase transition point in the chiral could be extremely strong. limit is well defined, while that of crossover does not has We first present the velocity vρ in the scenario of constant a strict definition; one of the usually used definitions is the mass in order to illustrate the influence of the phase transition maximum of susceptibility dmψ /dT. In the expanding sys- on the evolution of the system; see Fig. 6. We take constant tem, the phase transition point in the chiral limit can still be homogeneous mass mψ (τ,ρ) = 150 MeV and solve only the defined by the space-time point where the quark mass reaches transport equation for a free streaming system; the force term

014901-7 ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021)

FIG. 4. The evolution of quark mass mψ (τ,ρ) and transverse velocity vρ (τ,ρ) in the real case and the chiral limit, with a comparison between large and small relaxation times. The lines are rainbow colored, representing different evolution times; the evolution time τ is in units of fm/c, and the phase transition point is marked by the dotted line. vanishes since ∂τ m = 0 and ∂ρ m = 0. In this scenario, the are accelerated. In comparison, when the force term is con- kink does not appear, indicating that the kink arises from the sidered, the quarks near the phase boundary are slowed down. inhomogeneous mass distribution and thus the force term. The appearance of kinks is closely related to the phase tran- In order to further illustrate the influence of the force term, sition and the spatial distribution of the quark mass. The we now concurrently solve the transport equation and the velocity kinks are more obvious in scenarios with small cur- gap equation, but remove the force terms in the transport rent mass or small relaxation time. This phenomenon can equation by fixing ∂τ m = 0 and ∂ρ m = 0. The evolutions of be understood as follows: since the force term is related mass and thermodynamic quantities has no obvious difference to the susceptibility as well as the gradient of temperature compared to those of self-consistent solutions; however, the ∇m = (dm/dT)∇T , the phase transition of the chiral limit velocity vρ is quite different, which is presented as dashed has divergent susceptibility while that of the real case is finite, lines in Fig. 7, with the solid lines being the velocity of the thus the kinks in the chiral limit panels are more obvious self-consistent solution. As shown in each panel of Fig. 7, compared with those in the real case panels. The expansion when the force terms are ignored, the velocity has a bump starts with equilibrium distribution and gradually becomes around the phase transition. Namely, if there is phase transi- out of equilibrium due to the huge pressure gradient. With tion but no force term, the quarks near the phase boundary large relaxation time, the system spends longer time in the

014901-8 CHIRAL PHASE TRANSITION IN AN EXPANDING QUARK … PHYSICAL REVIEW C 103, 014901 (2021)

FIG. 5. The phase transition hypersurface obtained by self- consistent solution of a chiral limit system. Different lines represent different relaxation times. out-of-equilibrium state, and the critical effect is further FIG. 7. The transverse velocity vρ for the real case and the chiral washed out. limit, with different relaxation times. The solid lines are transverse velocity vρ obtained by self-consistent solution. The dashed lines are transverse velocity vρ obtained by solving together the transport D. Distribution function equation and gap equation, but ignoring the force terms in the trans- The phase transition hypersurface affects the motion of port equation. The lines are rainbow colored, representing different τ / quarks around it, and has an influence on the pT spectrum times in the evolution, and the evolution time is in units of fm c. of the quarks in the system. This can be directly revealed from the distribution function. Under the given symmetry ponent a , which is obtained by integrating the distribution + 0 in Sec. III, the distribution function is defined on a (4 1)- function f (ρ, pT ,θ,ξ) over θ, is related to the density dis- ρ, p⊥,θ,ξ τ dimensional phase space, ( )aswellas , where tribution. The first component a1 obtained by integrating θ is the angle between the transverse coordinate ρ and f (ρ, p ,θ,ξ) cos θ over θ gives a hint to the direction of the θ T the transverse momentum pT .The dependence has been particle velocity. expanded to a series of Fourier coefficients. The zeroth com- In Fig. 8, we present the zeroth and the first Fourier components in the scenario with weak collisions, namely with large relaxation time τθ = 10/Teq. The solid lines correspond to the self-consistent distribution function, while the dashed lines are the distribution function where the force term is ignored. Coefficients a0 and a1 as a function of transverse momentum pT at various given transverse coordinates ρ/ρ0 are presented in the left panel. The right panel shows the coefficients a0 and a1 as a function of transverse coordinates ρ/ρ0 at various given transverse momenta pT . At some evolution time τ = 1.0fmor(τ − τ0 )/τ0 = 1, the phase transition takes place around the position ρ/ρ0 ≈ 0.7, which is also the location of the kink in the velocity; see Fig. 4. From the left panel of Fig. 8, in the self-consistent solution (solid lines), the distribution function a1 is negative at low pT for ρ/ρ0 around 0.6 to 0.8, which means the particles with low momentum are bounced back by the phase transition “wall,” while the dashed lines reveal that, without the force term, the particles cannot see the “wall.” This effect is also obvious from the a1 in the right panel where, for small momenta, FIG. 6. Velocity in the ρ direction of the expansion with constant the distribution function changes sign around ρ/ρ0 = 0.7; mass mψ (τ,ρ) = 150 MeV and infinite relaxation time. The lines are for large momenta, the distribution function stays positive rainbow colored, representing different times in the evolution, and but has smaller values. This indicates that the particles with the evolution time τ is in units of fm/c. small momentum are bounced back by the phase transition

014901-9 ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021)

FIG. 8. Zeroth ande ﬁrst Fourier components in the case where the collisions are weak, τθ = 10/Teq; the solid lines are the distribution functions from the self-consistent solution, the dashed lines are the distribution functions when the force term is ignored. The left panel shows a0 and a1 as functions of transverse momentum pT at various given transverse coordinates ρ/ρ0, the right panel shows a0 and a1 as functions of transverse coordinates ρ/ρ0 at various given transverse momenta pT .

wall around ρ/ρ0 = 0.7; particles with large momentum go system. This requires understanding of the chiral phase tran- through the “wall,” but have been slowed down. The integral sition in an expanding, out-of-equilibrium system. of a0(ρ, pT ,ξ) over pT at fixed ρ corresponds to the num- In this work, we investigate the evolution of an expanding ber density of particles somewhere in the transverse plane, quark-antiquark system, self-consistently taking into account while the integral of a0(ρ, pT ,ξ) over ρ at fixed pT is the the dynamical quark constituent mass. In order to reduce the number density of particles of some fixed momentum. It can dimension of the phase space, we consider a longitudinal be observed from the left panel, whether the force term is boost-invariant and transversal rotational-symmetric system, ignored or not, that the number density away from the “wall” which is a good approximation for ultracentral heavy ion (ρ/ρ0 = 0.7) is similar, while the force term would collect collisions. In the numerical process, both Vlasov and gap more particles around the “wall.” From the right panel, there equations are solved concurrently, giving a self-consistent are more low momentum particles kept inside the wall be- evolution of both the quark-antiquark distribution function cause of the force term. and the quark constituent mass. The space-time-dependent As is presented in Fig. 9, the effect of the phase transition constituent mass serves as the chiral order parameter and “wall” is smoothed by the collision. Since the particles are affects the evolution of the quark distribution function through relaxed into thermal distribution, the direction of a single the force term. In order to investigate the off-equilibrium particle is aligned along the collective velocity. With strong effects, we introduce the relaxation time approximation for the collisions, the influence of the force term on number density collision term, and compare the local equilibrium and out-of- almost disappears, giving the same a0 whether or not the equilibrium results by considering small and large relaxation force term is taken into consideration. The first-order compo- times. nents a1 are still affected by the force term. Inside the “wall” The evolution of the quark mass illustrates the chiral phase (ρ/ρ0 < 0.7), the force term does not has any influence. Out- transition, and defines the phase diagram in the space-time. A side the “wall” (ρ/ρ0 > 0.7), the distribution a1 is smaller kink in the transverse velocity is observed around the phase when the force term is considered, which means the particles transition boundary, which appears because of the large force are slowed down by the force term. term around the phase transition. The kink is more obvious for smaller current mass and smaller relaxation time, which means that the crossover and nonequilibrium effect tend to V. SUMMARY smooth out the kink. The influence of the phase transition Quite different from the equilibrium thermodynamics, the hypersurface is further investigated by directly analyzing the realistic chiral phase transition in a heavy ion collision takes distribution function. It is observed that the phase transition place in a highly inhomogeneous, fast evolving dynamical wall would bounce back the low momentum particles, thus

014901-10 CHIRAL PHASE TRANSITION IN AN EXPANDING QUARK … PHYSICAL REVIEW C 103, 014901 (2021)

FIG. 9. Zeroth and ﬁrst Fourier components in the case where the collisions are strong, τθ = 0.005/Teq; the solid lines are the distribution functions from the self-consistent solution, the dashed lines are the distribution functions when the force term is ignored. The left panel shows a0 and a1 as functions of transverse momentum pT at various given transverse coordinates ρ/ρ0, the right panel shows a0 and a1 as functions of transverse coordinates ρ/ρ0 at various given transverse momenta pT .

giving a kink in the velocity, and may enhance the low pT eter at the first-order phase transition at large density and the part of the momentum spectrum. very strong dynamical fluctuations around the critical point at To see clearly the effect of the chiral phase transition in finite baryon density, the chiral phase transition induced kink the final state of heavy-ion collisions is still an open question. and low pT enhancement may lead to a sizable effect on the Since there is no strict chiral phase transition in the formed final state hadron distributions. Such a study is in progress. hot medium, the kink and low pT enhancement induced by the chiral crossover are not so strong and hard to be clearly ACKNOWLEDGMENTS measured in the final state. A possible candidate to distinctly signal the chiral phase transition is the direct flow v1, which is The work is supported by the NSFC Grants No. 11890712 more sensitive to the pressure of the expansion and has been and No. 12005112, the Postdoctoral Innovative Talent Support treated as a signal of phase transitions in high energy nuclear Program of China. S.S. is supported by the Natural Sciences collisions [37,38]. Another way to clearly see the effect of the and Engineering Research Council of Canada, and the Fonds chiral phase transition is probably the quark matter at finite de recherche du Québec - Nature et technologies (FRQNT) baryon density, which can be realized in intermediate energy through the Programmede Bourses dExcellencepour Étudi- nuclear collisions. Considering the jump of the order param- ants Étrangers (PBEEE).

[1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, The [4] P. de Forcrand, Simulating QCD at finite density, PoS Order of the quantum chromodynamics transition predicted by LAT2009, 010 (2009). the standard model of particle physics, Nature (London) 443, [5] O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H. Rischke, 675 (2006). Chiral phase transition within effective models with constituent [2] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. quarks, Phys.Rev.C64, 045202 (2001). Ratti, and K. K. Szabo, Is there still any Tc mystery in lattice [6] B.-J. Schaefer, J. M. Pawlowski, and J. Wambach, Phase struc- QCD? Results with physical masses in the continuum limit III, ture of the Polyakov-quark-meson model, Phys.Rev.D76, J. High Energy Phys. 09 (2010) 073. 074023 (2007). [3] Z. Fodor and S. D. Katz, A New method to study lattice QCD [7] K. Fukushima, Phase diagrams in the three-flavor Nambu–Jona- at finite temperature and chemical potential, Phys. Lett. B 534, Lasinio model with the Polyakov loop, Phys. Rev. D 77, 114028 87 (2002). (2008).

014901-11 ZIYUE WANG, SHUZHE SHI, AND PENGFEI ZHUANG PHYSICAL REVIEW C 103, 014901 (2021)

[8] T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, The phase [23] A. Meistrenko, C. Wesp, H. van Hees, and C. Greiner, Nonequi- structure of the Polyakov-quark-meson model beyond mean librium dynamics and transport near the chiral phase transition field, Phys. Lett. B 696, 58 (2011). of a quark-meson model, J. Phys. Conf. Ser. 503, 012003 [9] M. M. Aggarwal et al., Higher Moments of Net-Proton Mul- (2014). tiplicity Distributions at RHIC, Phys.Rev.Lett.105, 022302 [24] C. Wesp, H. van Hees, A. Meistrenko, and C. Greiner, Kinetics (2010). of the chiral phase transition in a linear σ model, Eur. Phys. J. [10] X. Luo, Search for the QCD critical point by higher moments A 54, 24 (2018). of net-proton multiplicity distributions at STAR, Nucl. Phys. A [25] Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elemen- 904-905, 911c (2013). tary Particles Based on an Analogy with Superconductivity. 1, [11] L. Adamczyk et al., Energy Dependence of Moments of Net- Phys. Rev. 122, 345 (1961). Proton Multiplicity Distributions at RHIC, Phys. Rev. Lett. 112, [26] S. P. Klevansky, The Nambu–Jona-Lasinio model of quantum 032302 (2014). chromodynamics, Rev. Mod. Phys. 64, 649 (1992). [12] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Signatures [27] T. Hatsuda and T. Kunihiro, QCD phenomenology based on a of the Tricritical Point in QCD, Phys.Rev.Lett.81, 4816 chiral effective Lagrangian, Phys. Rep. 247, 221 (1994). (1998). [28] D. Vasak, M. Gyulassy, and H. T. Elze, Quantum transport [13] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Event-by- theory for Abelian plasmas, Ann. Phys. (NY) 173, 462 (1987). event fluctuations in heavy ion collisions and the QCD critical [29] I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, Phase space point, Phys.Rev.D60, 114028 (1999). structure of the Dirac vacuum, Phys.Rev.D44, 1825 (1991). [14] M. A. Stephanov, Non-Gaussian Fluctuations Near the QCD [30] P. Zhuang and U. W. Heinz, Relativistic quantum transport Critical Point, Phys.Rev.Lett.102, 032301 (2009). theory for electrodynamics, Ann. Phys. (NY) 245, 311 (1996). [15] M. A. Stephanov, Sign of Kurtosis near the QCD Critical Point, [31] P.-F. Zhuang and U. W. Heinz, Relativistic kinetic equations Phys.Rev.Lett.107, 052301 (2011). for electromagnetic, scalar and pseudoscalar interactions, Phys. [16] X. Luo and N. Xu, Search for the QCD critical point with Rev. D 53, 2096 (1996). fluctuations of conserved quantities in relativistic heavy-ion [32] P.-F. Zhuang and U. W. Heinz, Equal-time hierarchies in quan- collisions at RHIC: An overview, Nucl. Sci. Technol. 28, 112 tum transport theory, Phys. Rev. D 57, 6525 (1998). (2017). [33] X. Guo and P. Zhuang, Out-of-equilibrium UA(1) symmetry [17] B. Berdnikov and K. Rajagopal, Slowing out of equilib- breaking in electromagnetic fields, Phys. Rev. D 98, 016007 rium near the QCD critical point, Phys. Rev. D 61, 105017 (2018). (2000). [34] G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kofman, A. D. [18] A. Abada and J. Aichelin, Chiral Phase Transition in an Linde, and I. Tkachev, Dynamics of Symmetry Breaking and Expanding Quark Gluon Plasma, Phys.Rev.Lett.74, 3130 Ttachyonic Preheating, Phys.Rev.Lett.87, 011601 (2001). (1995). [35] F. Cooper, Y. Kluger, E. Mottola, and J. P. Paz, Nonequilibrium [19] A. Abada and M. C. Birse, Coherent amplification of classical quantum dynamics of disoriented chiral condensates, Phys. Rev. pion fields during the cooling of droplets of quark plasma, Phys. D 51, 2377 (1995). Rev. D 55, 6887 (1997). [36] S. Gavin, A. Gocksch, and R. D. Pisarski, How to Make Large [20] C. Greiner and D.-H. Rischke, Shell-like structures in an ex- Domains of Disoriented Chiral Condensate, Phys. Rev. Lett. 72, panding quark-antiquark plasma, Phys. Rev. C 54, 1360 (1996). 2143 (1994). [21] Z.-W. Yang and P.-F. Zhuang, Deconfinement phase transition [37] Y. Nara, H. Niemi, A. Ohnishi, and H. Stöcker, Examination of in an expanding quark system in relaxation time approximation, directed flow as a signature of the softest point of the equation Phys.Rev.C69, 035203 (2004). of state in QCD matter, Phys. Rev. C 94, 034906 (2016). [22] H. van Hees, C. Wesp, A. Meistrenko, and C. Greiner, Dynam- [38] L. Adamczyk et al., Beam-Energy Dependence of Directed ¯ ± 0 φ ics of the chiral phase transition, Acta Phys. Pol. Suppl. 7,59 Flow of , , K , Ks ,and in Au+Au Collisions, Phys. Rev. (2014). Lett. 120, 062301 (2018).

014901-12