An effective thermodynamic potential from the instanton with Polyakov-loop contributions

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We derive an effective thermodynamic potential (Omegaₑff) at finite temperature (T>0) and zero quark-chemical potential (muᵣ=0), using the singular-gauge instanton solution and Matsubara formula for Nc=3 and Nf=2 in the chiral limit. The momentum-dependent constituent-quark mass is also obtained as a function of T, employing the Harrington-Shepard caloron solution in the large-Nc limit. In addition, we take into account the imaginary quark chemical potential muᵢ = A₄, translated as the traced Polayakov-loop (Phi) as an order parameter for the Z(Nc) symmsetry, characterizing the confinement (intact) and deconfinement (spontaneously broken) phases. As a result, we observe the crossover of the chiral (chi) order parameter sigma² and Phi. It also turns out that the critical temperature for the deconfinment phase transition, Tᶻc is lowered by about (5-10)% in comparison to the case with a constant constituent-quark mass. This behavior can be understood by considerable effects from the partial chiral restoration and nontrivial QCD vacuum on Phi. Numerical calculations show that the crossover transitions occur at (Tᶜhic,Tᶻc) (216,227) MeV.