ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 9, с. 1189-1203
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ DYNAMICS AND STABILITY OF CHIRAL FLUID
©2014 I. N. Mishustin1),2), T. Koide3),4), G. S. Denicol3),5), G. Torrieri1),6)
Received September24, 2013; in final form, January 9, 2014
Starting from the linear sigma model with constituent quarks we derive hydrodynamic equations which are coupled to the order-parameter field, e.g. the chiral fluid dynamics. For a static system in thermal equilibrium this model leads to a chiral phase transition which, depending on the choice of the quark—meson coupling constant g, could be a crossover or a first order one. We investigate the stability of the chiral fluid in the static and expanding background by considering the evolution of perturbations with respect to the mean-field solution. In the static background the spectrum of plane-wave perturbations consists of two branches, one corresponding to the sound waves and another to the a-meson excitations. For large g these two branches cross and the excitation spectrum acquires exponentially growing modes. The stability analysis is also done for the Bjorken-like background solution by explicitly solving the time-dependent differential equation for perturbations in the n space. In this case the growth rate of unstable modes is significantly reduced.
DOI: 10.7868/S0044002714090074
1. INTRODUCTION
Creation of new forms of strongly interacting matter, in particular, the observation of a deconfined and chirally restored state, is the main goal of present and future experiments with relativistic heavy-ion beams [1—5]. Significant progress in understanding the dynamics of relativistic heavy-ion collisions and the properties of dense matter produced in such collisions have been achieved in experiments at Rel-ativistic Heavy Ion Collider (RHIC) [6—9]. A few years ago the Large Hadron Collider (LHC) has started a new era in high-energy nuclear physics by colliding p + p and Au + Au beams at much higher energies than in previous experiments, and new interesting results have been obtained already, see, e.g., [10] and references therein. Investigations of the phase diagram of strongly interacting matter and, in particular, searching for manifestations of the QCD-based phase transitions remain in the focus of theoretical and experimental studies.
1)Frankfurt Institute for Advanced Studies, J.W. Goethe University, Frankfurt am Main, Germany.
2)National Research Centre "Kurchatov Institute", Moscow, Russia.
3)Institute for Theoretical Physics, J.W. Goethe University, Frankfurt am Main, Germany.
4)Instituto de Fisica, Universidade Federal do Rio de Janeiro, Brazil.
5)Department of Physics, McGill University, Montreal, Quebec, Canada.
6)Pupin Physics Laboratory, Columbia University, New York, USA.
The signatures of such phase transitions have been studied mostly based on equilibrium concepts. However, the process of a relativistic heavy-ion collision at RHIC and LHC energies is very fast and one should expect such a phase transition to be strongly affected by the dynamics. Thus, it is important to study the phase transition dynamics in a time-dependent background. Previously non-equilibrium effects associated with the chi-ral/deconfinement phase transition have been studied within several macroscopic approaches [11—24].
Generally, in order to investigate non-equilibrium effects, one should solve a quantum many-body problem by using, e.g. Kadanoff—Baym equation. But applying such formalism to relativistic heavy-ion collisions is very complicated, see, e.g., [25]. Fortunately, there is still hope that the collective behavior of the hot matter created in heavy-ion collisions can be described by a coarse-grained macroscopic theory, such as hydrodynamics. Indeed, some aspects of relativistic heavy-ion collisions, such as the collective flow of the produced matter, are well described in the framework of hydrodynamic models [26—34]. However, in order to describe a phase transition in a time-dependent background, the usual hydrodynamic model should be modified by explicitly considering the dynamics of the order parameter. This becomes necessary when the time scale associated with the order parameter relaxation becomes of the same order or longer than the characteristic time associated with the hydrodynamic variables. In the opposite limit, the effect of a phase transition can be
taken into account through the equation of state, as is usually done in macrophysics (see, e.g., [34]).
In this paper we use a modified hydrodynamic theory, namely the Chiral Fluid Dynamics (CDF), in which the fluid evolution is coupled to the dynamics of the chiral order parameter. This model was firstly proposed in [35] and further developed in several works [36—39]. Recently it was generalized by including fluctuations of the order parameter and dissipative terms [40—42]. It can be derived from the linear sigma model by assuming that microscopic and macroscopic degrees of freedom are clearly separated. Then the coarse-grained macroscopic dynamics is described only by a reduced number of variables, which are called the gross variables [43—45].
As was first pointed out by van Hove [46], the relaxation time of the order parameter increases near the second-order phase transition and it diverges at the critical point. This phenomenon is known as "critical slowing down" [47]. Its importance for modeling phase transitions in dynamical environments was demonstrated in [48]. In this situation non-equilibrium effects need to be considered explicitly even in an ideal fluid. This can be done by choosing the order parameter as another gross variable.
In the region of the phase diagram where the deconfinement/chiral phase transition is of first order, an extra time-scale appears, which is associated with the nucleation process [49]. Only when this time scale is short with respect to the hydrodynamic time scale, we can assume a two-phase equilibrium and the equation of state given by the Maxwell construction. However, if it is not the case, the fluid will pass through a metastable region of the equation of state and can even reach the point of spinodal instability [11 — 14, 24, 42]. In order to investigate this possibility, we study the stability of fluctuations around a hydrostatic state and a Bjorken-type expanding state. By comparing the results, we will be able to identify those features of the phase transition which are affected by the fast dynamics.
The paper is organized as follows. In Section 2 we derive basic equations of CFD from the linear a model with constituent quarks. In Section 3 we discuss the predictions of this model for the equilibrated system undergoing a chiral phase transition. Then in Section 4 we calculate the excitation spectrum of the system by introducing fluctuations around the static background solution. In Section 5 we study the time evolution of fluctuations in the Bjorken-like expanding background. Our concluding remarks are presented in Section 6.
2. DERIVATION OF CHIRAL FLUID DYNAMICS
As the low-energy effective theory of QCD, we adopt the linear sigma model with constituent quarks [50] whose qualitative features (chiral symmetry, universality class, phase transition structure) are thought to coincide with QCD [51, 52]. More recently the thermodynamics of this model was studied on the mean-field level [53], as well as including the field fluctuations [54—57]. Following the previous works [35, 36], we describe the coarsegrained dynamics of the quark degrees of freedom with the hydrodynamic variables, coupled to the order parameter field a via its equation of motion.
The Lagrangian of the linear a model is
L = dn - g(a + ij5rn))q +
(1)
1
+ -[(d,a)2 + (d,Tv)2]-V(a,ir),
where q is the quark field, a and n are the chiral fields, g is the quark—meson coupling constant. The "Mexican Hat" potential V is given by
A 2
V(a,7v) = T(a2 + (7v)2-v2)2-Ha, (2)
where A, v, and H are the model parameters. They can be calculated by using the physical pion mass mn = = 138 MeV, the pion decay constant fn = 93 MeV and the sigma mass in vacuum ml = 600 MeV: H = fnml, v2 = f2 - ml/A2, and A2 = (ml -— ml)/(2f2). This potential provides a mechanism for spontaneous breaking of the chiral symmetry with non-zero vacuum expectation value of the sigma field,
avac = fn.
The coupling constant g is usually chosen so that the constituent quark mass in vacuum, gavac, is equal to one-third of the vacuum nucleon mass, that is, g = 3.3. Then the chiral phase transition at the vanishing chemical potential is shown to be of the crossover type. But the phase diagram has a critical (end) point at a finite baryon chemical potential, where the order of the phase transition changes to first order [53]. In this paper we limit our consideration to the case of vanishing chemical potential. In this case the effect of the first-order phase transition can be studied by changing the magnitude of the coupling constant g, as was proposed in [15]. In order to study the dynamics of a first-order phase transition, we shall also consider the case of g = 4.5, in which the phase transition is of first order.
For the sake of simplicity, the pion field is neglected in the rest of this paper and the sigma field a is considered as a classical field (condensate). We consider an idealized situation where the quark degrees of freedom have already achieved local thermal
equilibrium and can be approximately described as an ideal fluid, characterized by the energy density e, the pressure P and the fluid four-velocity uOn the other hand, the sigma field behaves as a classical external field (the chiral order parameter) acting on quarks through the mass term M = go. Then the energy— momentum tensor of the system can be represented as
Tßv __I
1 ~ Tfluider 1 field-
(3)
The energy—momentum tensor of an ideal fluid is generally represented as
fld _ te + P )ußuv - Pg
ßv
(4)
where e and P are the proper scalar densiti
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