7 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 $^{∗}$: 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 with 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}$$_{0}$A$_{0}$ via the covariant derivative D$_{µ}$ = ∂$_{µ}$ − iA$_{µ}$ . The Φ is a complex 3 × 3 matrix and is defined in terms of the scalar σ$_{a}$ and pseudoscalar π$_{a}$ meson nonets, where we have separated the contribution of chiral de- grees of freedom and the Polyakov loop. The chiral part of the Lagrangian, L chiral = L q + L m consists of the fermionic part and the purely mesonic contribution The 3 × 3 matrix H breaks the symmetry explicitly and is chosen as The critical temperature T$_{0}$ for deconfinement in the pure gauge sector is fixed at 270 MeV, in agreement with the lattice results. we obtain the thermodynamical potential density as with the quarks and antiquarks contribution The quantity U ( φ, φ $^{∗}$, T ) is the Polyakov-loop effec- tive potential expressed by the dynamics of the traced Polyakov loop The Polyakov loop L is a matrix in color space and ex- plicitly given by with β = 1 /T being the inverse of temperature and A$_{4}$ = iA $^{0}$. In the Polyakov gauge, the Polyakov-loop ma- trix can be given as a diagonal representation [57]. The coupling between Polyakov loop and quarks is uniquely determined by the covariant derivative D$_{µ}$ in the PLSM U ( φ, φ $^{∗}$, T ) T 4 = − b$_{2}$ ( T ) 2 | φ | 2 − b$_{3}$ 6 ( φ 3 + φ ∗ $^{3}$) + b$_{4}$ 4 ( | φ | $^{2}$)$^{2}$, (28) b$_{2}$ ( T ) = a$_{0}$ + a$_{1}$ ( T$_{0}$ T ) + a$_{2}$ ( T$_{0}$ T ) 2 + a$_{3}$ ( T$_{0}$ T ) 3 . (29) L m = Tr( ∂$_{µ}$ Φ $^{†}$∂ $^{µ}$Φ − m $^{2}$Φ $^{†}$Φ) − λ$_{1}$ [Tr(Φ $^{†}$Φ)] 2 − λ$_{2}$ Tr(Φ $^{†}$Φ) 2 + c [Det(Φ) + Det(Φ $^{†}$)] +Tr[ H (Φ + Φ $^{†}$)] , (22) L q = ∑ f ψ$_{f}$ ( iγ $^{µ}$D$_{µ}$ − gT$_{a}$ ( σ$_{a}$ + iγ$_{5}$π$_{a}$ )) ψ$_{f}$ (21) Φ = T$_{a}$ ( σ$_{a}$ + iπ$_{a}$ ) . (23) H = T$_{a}$h$_{a}$, (24) f$_{ab}$$_{0}$ ≡ 0 , d$_{ab}$$_{0}$ = √ 2 3 δ$_{ab}$. (25) a$_{0}$ = 6 . 75 , a$_{1}$ = − 1 . 95 , a$_{2}$ = 2 . 625 , a$_{3}$ = − 7 . 44 , b$_{3}$ = 0 . 75 , b$_{4}$ = 7 . 5 . (30) Ω( T, µ$_{f}$ ) = − T ln Z V = U ( σ$_{x}$, σ$_{y}$ ) + U ( φ, φ $^{∗}$, T ) + Ω ¯ $_{ψψ}$, (31) Ω ¯ ψψ = − 2 T N$_{q}$ ∫ d $^{3}$⃗ p (2 π ) 3 { ln[1 + 3( φ + φ $^{∗}$e− ( E$_{q}$ − µ ) $^{/T}$) e − ( E$_{q}$ − µ ) /T + e − 3( E$_{q}$ − µ ) $^{/T}$] +ln[1 + 3( φ ∗ + φe − ( E$_{q}$ + µ ) $^{/T}$) e − ( E$_{q}$ + µ ) /T + e − 3( E$_{q}$ + µ ) $^{/T}$] } − 2 T N$_{s}$ ∫ d $^{3}$⃗ p (2 π ) 3 { ln[1 + 3( φ + φ $^{∗}$e− ( E$_{s}$ − µ ) $^{/T}$) e − ( E$_{s}$ − µ ) /T + e − 3( E$_{s}$ − µ ) $^{/T}$] +ln[1 + 3( φ ∗ + φe − ( E$_{s}$ + µ ) $^{/T}$) e − ( E$_{s}$ + µ ) /T + e − 3( E$_{s}$ + µ ) $^{/T}$] } . (32) L ( ⃗ x ) = P exp [ i ∫ β 0 dτA$_{4}$ ( ⃗ x, τ ) ] , (27) φ = (Tr$_{c}$ L ) /N$_{c}$, φ ∗ = (Tr$_{c}$ L $^{†}$) /N$_{c}$. (26) 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