Thermodynamic Potential of the Polyakov Loop in SU(3) Quenched Lattice QCD

Using SU(3) lattice QCD, we study for the first time the effective potential of the Polyakov loop $\langle P \rangle$ at finite temperature, i.e., the thermodynamic potential, in the field-theoretical way. In the framework of the reweighting method in lattice QCD, we express the effective potential $V_{\rm eff}(\langle P \rangle)$ using the expectation value without a source term. In particular, we consider the most difficult and interesting case of vacuum coexistence at the critical temperature $T_c$. We adopt SU(3) quenched lattice QCD on $48^3 \times 6$ at $\beta$= 5.89379, which corresponds exactly to the critical temperature $T_c$ of the deconfinement phase transition, and use 200,000 Monte Carlo configurations. After categorizing the gauge configurations into one $Z_3$-symmetric and three $Z_3$-broken vacua each, we perform a vacuum-associated reweighting method, using the gauge configurations around each vacuum separately. Finally, we obtain the Polyakov-loop effective potential, which is well depicted around the $Z_3$-symmetric and $Z_3$-broken vacua.