content_image/431296.jpg content_image/431295.jpg content_image/431304.jpg content_image/431303.jpg content_image/431297.jpg content_image/431298.jpg content_image/431299.jpg content_image/431293.jpg content_image/431294.jpg content_image/431300.jpg content_image/431301.jpg content_image/431302.jpg
where we have separated the contribution of chiral de- grees of freedom and the Polyakov loop. The chiral part of the Lagrangian, L chiral = L q + L m consists of the fermionic part 

<img src='content_image/431296.jpg'>

and the purely mesonic contribution 

<img src='content_image/431295.jpg'>

the sum is over the three flavors (f=1,2,3 for u, d, s). In the above equation we have introduced a flavor-blind Yukawa coupling g of the quarks to the mesons and the coupling of the quarks to a background gauge field A$_{µ}$ = δ$_{µ}$$_{0}$A$_{0}$ via the covariant derivative D$_{µ}$ = ∂$_{µ}$ − iA$_{µ}$ . The Φ is a complex 3 × 3 matrix and is defined in terms of the scalar σ$_{a}$ and pseudoscalar π$_{a}$ meson nonets, 

The 3 × 3 matrix H breaks the symmetry explicitly and is chosen as 

where h$_{a}$ are nine external fields. The T$_{a}$ = λ$_{a}$/ 2 are the generators of the U (3) symmetry, λ$_{a}$ are the Gell- Mann matrices with λ$_{0}$ = √ 2 $_{3}$1 . The T$_{a}$ are normalized to Tr( T$_{a}$T$_{b}$ ) = δ$_{ab}$/ 2 and obey the U (3) algebra with [ T$_{a}$, T$_{b}$ ] = if$_{abc}$T$_{c}$ and { T$_{a}$, T$_{b}$ } = d$_{abc}$T$_{c}$ respectively, here f$_{abc}$ and d$_{abc}$ for a, b, c = 1 , ..., 8 are the standard an- tisymmetric and symmetric structure constants of SU (3) group and 

The quantity U ( φ, φ $^{∗}$, T ) is the Polyakov-loop effec- tive potential expressed by the dynamics of the traced Polyakov loop 

The Polyakov loop L is a matrix in color space and ex- plicitly given by 

with β = 1 /T being the inverse of temperature and A$_{4}$ = iA $^{0}$. In the Polyakov gauge, the Polyakov-loop ma- trix can be given as a diagonal representation [57]. The coupling between Polyakov loop and quarks is uniquely determined by the covariant derivative D$_{µ}$ in the PLSM 

<img src='content_image/431304.jpg'>

<img src='content_image/431303.jpg'>

<img src='content_image/431297.jpg'>

<img src='content_image/431298.jpg'>

<img src='content_image/431299.jpg'>

Lagrangian in Eq.(20), and in the chiral limit, this La- grangian is invariant under the chiral flavor group, just like the original QCD Lagrangian. The trace of the Polyakov-loop, φ and its conjugate φ ∗ can be treated as classical field variables in this work. 

The temperature dependent effective potential U ( φ, φ $^{∗}$, T ) is used to reproduce the thermodynamical behavior of the Polyakov loop for the pure gauge case in accordance with lattice QCD data, and it has the Z (3) center symmetry like the pure gauge QCD Lagrangian. In the absence of quarks, we have φ = φ ∗ and the Polyakov loop is taken as an order parameter for deconfinement. For low temperatures, U has a single minimum at φ = 0, while at high temperatures it develops a second one which turns into the absolute minimum above a critical temperature T$_{0}$ , and the Z (3) center symmetry is spontaneously broken. In this paper, we will use the potential U ( φ, φ $^{∗}$, T ) proposed in Ref.[58], which has a polynomial expansion in φ and φ $^{∗}$: 

<img src='content_image/431293.jpg'>

with 

<img src='content_image/431294.jpg'>

A precision fit of the constants a$_{i}$, b$_{i}$ is performed to re- produce the lattice data for pure gauge theory thermo- dynamics and the behavior of the Polyakov loop as a function of temperature. The corresponding parameters are 

<img src='content_image/431300.jpg'>

The critical temperature T$_{0}$ for deconfinement in the pure gauge sector is fixed at 270 MeV, in agreement with the lattice results. 

we obtain the thermodynamical potential density as 

<img src='content_image/431301.jpg'>

with the quarks and antiquarks contribution 

<img src='content_image/431302.jpg'>

