Advances in High Energy Physics

Volume 2015 (2015), Article ID 563428, 15 pages

http://dx.doi.org/10.1155/2015/563428

## On SU(3) Effective Models and Chiral Phase Transition

^{1}Egyptian Center for Theoretical Physics (ECTP), Modern University for Technology and Information (MTI), Cairo 11571, Egypt^{2}World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo 11571, Egypt^{3}Department of Physics, Brookhaven National Laboratory (BNL), P.O. Box 5000, Upton, NY 11973-5000, USA

Received 29 April 2015; Revised 15 August 2015; Accepted 6 September 2015

Academic Editor: Enrico Lunghi

Copyright © 2015 Abdel Nasser Tawfik and Niseem Magdy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Sensitivity of Polyakov Nambu-Jona-Lasinio (PNJL) model and Polyakov linear sigma-model (PLSM) has been utilized in studying QCD phase-diagram. From quasi-particle model (QPM) a gluonic sector is integrated into LSM. The hadron resonance gas (HRG) model is used in calculating the thermal and dense dependence of quark-antiquark condensate. We review these four models with respect to their descriptions for the chiral phase transition. We analyze the chiral order parameter, normalized net-strange condensate, and chiral phase-diagram and compare the results with recent lattice calculations. We find that PLSM chiral boundary is located in upper band of the lattice QCD calculations and agree well with the freeze-out results deduced from various high-energy experiments and thermal models. Also, we find that the chiral temperature calculated from HRG is larger than that from PLSM. This is also larger than the freeze-out temperatures calculated in lattice QCD and deduced from experiments and thermal models. The corresponding temperature and chemical potential are very similar to that of PLSM. Although the results from PNJL and QLSM keep the same behavior, their chiral temperature is higher than that of PLSM and HRG. This might be interpreted due the very heavy quark masses implemented in both models.

#### 1. Introduction

At large momentum scale, quantum chromodynamics (QCD) predicts asymptotic freedom [1, 2] or a remarkable weakening in the running strong coupling. Accordingly, phase transition takes place from hadrons in which quarks and gluons are conjectured to remain confined (at low temperature and density) to quark-gluon plasma (QGP) [3, 4], in which quarks and gluons become deconfined (at high temperature and density) [5]. Furthermore, at low temperature, the QCD chiral symmetry is spontaneously broken; . In this limit, the chiral condensate remains finite below the critical temperature (). The broken chiral symmetry is restored at high temperatures. The finite quark masses explicitly break QCD chiral symmetry.

Nambu-Jona-Lasinio (NJL) model [6] describes well the hadronic degrees of freedom. Polyakov Nambu-Jona-Lasinio (PNJL) model [7–9] takes into consideration the quark dynamics [10] and has been utilized to study the QCD phase-diagram [11, 12]. Also, linear- model (LSM) [13] can be used in mapping out the QCD phase-diagram.

Many studies have been performed on LSM like LSM [13] at vanishing temperature, LSM at finite temperature [14, 15], and LSM for , and quark flavors [16–19]. In order to obtain reliable results, extended LSM to PLSM can be utilized, in which information about the confining glue sector of the theory is included in form of Polyakov loop potential. The latter can be extracted from pure Yang-Mills lattice simulations [20–23]. Also, the Polyakov linear sigma-model (PLSM) and Polyakov quark meson model (PQM) [24–26] deliver reliable results. Furthermore, the quasi-particle model (QPM) [27, 28] was suggested to reproduce the lattice QCD calculations [29, 30], in which two types of actions are implemented; the lattice QCD simulations utilizing the standard Wilson action and the ones with renormalization improved action.

In the present work, we integrate the gluonic sector of QPM into LSM [31] (QLSM) in order to reproduce the recent lattice QCD calculations [32]. In QLSM [31], the Polyakov contributions to the gluonic interactions and to the confinement-deconfinement phase transition are entirely excluded. Instead we just add the gluonic part of QPM. Therefore, the quark masses should be very heavy. We will comment on this, later on. In Section 3, we outline the QLSM results. They are similar to that of PNJL. This might be interpreted due the very heavy quark masses implemented in both models. Similar approach has been introduced in [33]. The authors described inclusion of gluonic Polyakov loop, which is assumed to generate a large gauge invariance and lead to a remarkable modification in hadron thermodynamics. A quite remarkable bridging between PNJL model quantum and local Polyakov loop and HRG model has been introduced [34]. A large suppression of the thermal effects has been reported and it was concluded that the center symmetry breaking becomes exponentially small with increasing the masses of constituent quarks. In other words, the chiral symmetry restoration becomes exponentially small with increasing the pion mass.

The hadron resonance gas (HRG) model gives a good description for the thermal and dense evolution of various thermodynamic quantities in the hadronic matter [35–43]. Also, it has been successfully utilized to characterize the conditions deriving the chemical freeze-out at finite densities [44]. In light of this, the HRG model can be well used in calculating the thermal and dense dependence of quark-antiquark condensate [45]. The HRG grand canonical ensemble includes two important features [38]: the kinetic energies and the summation over all degrees of freedom and energies of the resonances. On the other hand, it is known that the formation of resonances can only be achieved through strong interactions [46]; resonances (fireballs) are composed of further resonances (fireballs), which in turn consist of resonances (fireballs) and so on. In other words, the contributions of the hadron resonances to the partition function are the same as that of free (noninteracting) particles with an effective mass. At temperatures comparable to the resonance half-width, the effective mass approaches the physical one [38]. Thus, at high temperatures, the strong interactions are conjectured to be taken into consideration through the inclusion of heavy resonances. It is found that the hadron resonances with masses up to 2 GeV are representing suitable constituents for the partition function [35–43]. In such a way, the singularity expected at the Hagedorn temperature [35, 36] can be avoided and the strong interactions are assumed to be taken into consideration. Nevertheless, validity of the HRG model is limited to the temperatures below the critical one, .

In the present paper, we review PLSM, QLSM, PNJL, and HRG with respect to their descriptions for the chiral phase transition. We analyse the chiral order-parameter , the normalized net-strange condensate , and the chiral phase-diagram and compare the results with the lattice QCD [47–49]. The present work is organized as follows. In Section 2, we introduce the different approaches PLSM [50] (Section 2.1), QLSM (Section 2.2), PNJL (Section 2.3), and HRG (Section 2.4). The corresponding mean field approximations are also outlined. Section 3 is devoted to the results. The conclusions and outlook will be given in Section 4.

#### 2. SU(3) Effective Models

##### 2.1. Polyakov Linear Sigma-Model (PLSM)

As discussed in [31, 50], the Lagrangian of LSM with quark flavors and (for quarks, only) color degrees and with quarks coupled to Polyakov loop dynamics was introduced in [26, 51]where the chiral part of the Lagrangian of the symmetric linear sigma-model Lagrangian with is [52, 53] . The first term is fermionic part (2) with a flavor-blind Yukawa coupling of the quarks. The second term is mesonic contribution (3):The summation runs over the three flavors ( for the three quarks , and ). The flavor-blind Yukawa coupling should couple the quarks to the mesons. The coupling of the quarks to the Euclidean gauge field is given via the covariant derivative [20, 21]. In (3), is a complex matrix which depends on the and [53], where are Dirac matrices, are the scalar mesons, and are the pseudoscalar mesons:where with are the nine generators of the symmetry group and are the eight Gell-Mann matrices [13]. The chiral symmetry is explicitly broken throughwhich is a matrix with nine parameters . Three finite condensates , , and are possible, because the finite values of vacuum expectation of and are conjectured to carry the vacuum quantum numbers and the diagonal components of the explicit symmetry breaking term, , where , and , and squared tree level mass of the mesonic fields , two possible coupling constants and , Yukawa coupling , and a cubic coupling constant can be estimated as follows: MeV, , and and .

In order to get a good analysis it is more convenient to convert the condensates and into a pure nonstrange and strange condensates [54]:

The second term in (1), , represents Polyakov loop effective potential [20], which agrees well with the nonperturbative lattice QCD simulations and should have center symmetry as pure gauge QCD Lagrangian does [8, 24]. In the present work, we use the potential as a polynomial expansion in and [8, 9, 55, 56]:where

In order to reproduce pure gauge lattice QCD thermodynamics and the behavior of the Polyakov loop as a function of temperature, we use the parameters , , and . For a much better agreement with the lattice QCD results, the deconfinement temperature in pure gauge sector is fixed at MeV.

###### 2.1.1. Polyakov Linear Sigma-Model (PLSM) in Mean Field Approximation

In thermal equilibrium the grand partition function can be defined by using a path integral over the quark, antiquark, and meson fields:where and is the volume of the system. is the chemical potential for . We consider symmetric quark matter and define a uniform blind chemical potential . Then, we evaluate the partition function in the mean field approximation [53, 57]. We can use standard methods [58] in order to calculate the integration. This gives the effective potential for the mesons.

We define the thermodynamic potential density of PLSM asAssuming degenerate light quarks, that is, , the quarks and antiquarks contribution potential is given as [51]where , and the valence quark and antiquark energy for light and strange quark are as follows: and , respectively. Also, as per [54] the light quark sector (13) decouples from the strange quark sector () and light quark mass gets simplified in this new basis to

The purely mesonic potential is given asWe notice that the sum in (7), (11), and (14) gives the thermodynamic potential density similar to (10), which has seven parameters , and , two unknown condensates and , and two order parameters for the deconfinement and . The six parameters , and are fixed in the vacuum by six experimentally known quantities [53]. In order to evaluate the unknown parameters , and , we minimize the thermodynamic potential (10) with respect to , and , respectively. Doing this, we obtain a set of four equations of motion:where min means , and are the global minimum:Accordingly, the chiral order parameter can be deduced as

##### 2.2. Linear Sigma-Model and Quasi-Particle Sector (QLSM)

When the Polyakov contributions to the gluonic interactions and to the confinement-deconfinement phase transition are entirely excluded, the Lagrangian of LSM with quark flavors and (for quarks, only) color degrees of freedom, where the quarks couple to the Polyakov loop dynamics, has been introduced in [26, 51]The main original proposal of the present work is the modification of (18):where the chiral part of the Lagrangian is of symmetry [52, 53]. Instead of , the gluonic potential , which is similar to the gluonic sector of the quasi-particle model, is inserted (review (33)). The Lagrangian with consists of two parts: fermionic and mesonic contributions (2) and (3), respectively.

Some details about the quasi-particle model are in order. The model gives a good phenomenological description for lattice QCD simulation and treats the interacting massless quarks and gluons as noninteracting massive quasi-particles [59]. The corresponding degrees of freedom are treated in a similar way as the electrons in condensed matter theory [60]; that is, the interaction with the medium provides the quasi-particles with dynamical masses. Consequently, most of the interactions can be taken into account. When confronting it to the lattice QCD calculations, the free parameters can be fixed. The pressure at finite and is given as where the function stands for bag pressure at finite and which can be determined by thermodynamic self-consistency and ; the stability of with respect to the self-energies () and the distribution function for bosons and fermions, , respectively, is given as The quasi-particle dispersion relation can be approximated by the asymptotic mass shell expression near the light cone [27, 28]: where is the self-energy at finite and and is a factor taking into account the mass scaling as used in the lattice QCD simulations. In other words, was useful when the lattice QCD simulations have been performed with quark masses heavier than the physical ones. In the present work, the gluon self-energies are relevant [61]: where the effective coupling at vanishing chemical potential is given as And the two-loop effective coupling reads [27] and is a regulator at . The parameter is used to adjust the scale as found in lattice QCD simulations. These two parameters are not very crucial in the present calculations. The regulator and scale are controlled by the condensates ( and ) and the order parameters ( and ), which are given as function of temperature and baryon chemical potential. The function [62] depends on the QCD coupling , , with being the energy scale. It is obvious that the QCD coupling in (24) and (25) is given in -dependence, only. In calculating at finite it is apparently needed to extend to be -dependent, as well. The two-loop perturbation estimation for functions gives

###### 2.2.1. Linear Sigma-Model and Quasi-Particle Sector (QLSM) in Mean Field Approximation

As in Section 2.1.1 and (9), we derive the thermodynamic potential density in the mean field approximation. This consists of three parts: mesonic and quasi-gluonic potentials in additional to the quark potential:(i)First, the quark potential part [53] isIt is obvious that is equivalent to . The occupation quark/antiquark numbers readand antiquarks , respectively. The number of internal quark degrees of freedom is denoted by and (for quarks and antiquarks). The energies are given aswith the quark masses which is related to and for -, and -, and -quarks, respectively. As given, the latter are proportional to the -fields:where the Yukawa coupling . The symbols for the chiral condensates, and for light- and strange-quarks, respectively, are kept as in the literature.(ii)Second, the purely mesonic potential part reads(iii)Third, the quasi-gluonic potential part is constructed from (22), (21), and (20): In (33), the degeneracy factor and two parameters and , which were given in (25), should be fixed in order to reproduce the lattice QCD calculations. Here, we find that and MeV give excellent results.

When adding the three potentials given in (33), (32), and (28), the thermodynamics and chiral phase translation can be analysed. The resulting potential can be used to determine the normalized net-strange condensate and chiral order parameter (51):where is function of and the quark masses should be very heavy. The QLSM results are similar to that of PNJL, Section 3. This might be interpreted due the very heavy quark masses implemented in both models.

##### 2.3. Polyakov Nambu-Jona-Lasinio (PNJL) Model

The Lagrangian of PNJL reads [63, 64]where the matrices are chiral projectors, is the Polyakov loop potential (Landau-Ginzburg type potential), and stand for gauge field interactions. The mass of a particular flavor is denoted by , where . The two coupling constants and , , are Gell-Mann matrices [13] and are Dirac matrices. The model is not renormalizable so that we have to use three-momentum cutoff regulator in order to keep quark loops finite.

The Polyakov loop potential is given by [65]with and being constants, and we choose the following fitting values for the potential parameters: , and MeV. These are adjusted to the pure gauge lattice data such that the equation of state and the Polyakov loop expectation values are reproduced.

###### 2.3.1. Polyakov Nambu-Jona-Lasinio (PNJL) Model in Mean Field Approximation

The thermodynamic potential density of PNJL is defined aswhere is the single quasi-particle energy, , and from isospin symmetry, . In the above integrals, the vacuum integral has a cutoff whereas the medium dependent integrals have been extended to infinity. By the self-consistent gap equation, the quark mass can be estimated:where denotes the chiral condensate of quark with flavor and other parameters are listed out in Table 1 [63, 64]. For isospin symmetry, we define the light and strange-quark masses asHere, we notice the strong dependence on the -fields.