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Confinement--deconfinement phase transition and gauge-invariant gluonic mass in Yang-Mills theory

We give an analytical derivation of the confinement/deconfinement phase transition at finite temperature in the $SU(N)$ Yang-Mills theory in the $D$-dimensional space time for $D>2$. We elucidate what is the mechanism for quark confinement and deconfinement at finite temperature and why the phase transition occurs at a certain temperature. For this purpose, we use a novel reformulation of the Yang-Mills theory which allows the gauge-invariant gluonic mass term and calculate analytically the effective potential of the Polyakov loop average concretely for the $SU(2)$ and $SU(3)$ Yang-Mills theories by including the gauge-invariant dynamical gluonic mass. For $D=4$, we give an estimate on the transition temperature $T_d$ as the ratio to the gauge-invariant gluonic mass $M$ which has been measured on the lattice at zero temperature and is calculable also at finite temperature. We show that the order of the phase transition at $T_d$ is the second order for $SU(2)$ and weakly first order for $SU(3)$ Yang-Mills theory. These initial results are obtained easily based on the analytical calculations of the "one-loop type" in the first approximation. Then these results are identified with the initial condition in solving the flow equation of the Wetterich type to improve the initial results in a systematic way in the framework of the functional renormalization group. But the improvements do not change the initial results in an essential way except for some thermodynamic observables. We argue how the artifacts in the first approximation are eliminated to obtain the correct behaviors for such thermodynamic observables.