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(2021) Chiral Phase Transition in an Expanding Quark System

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(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.
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