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We give an analytical derivation of the confinement/deconfinement phasetransition at finite temperature in the $SU(N)$ Yang-Mills theory in the$D$-dimensional space time for $D>2$. For this purpose, we use a novelreformulation of the Yang-Mills theory which allows the gauge-invariant gluonicmass term, and calculate analytically the effective potential of the Polyakovloop average concretely for the $SU(2)$ and $SU(3)$ Yang-Mills theories byincluding the gauge-invariant dynamical gluonic mass $M$. For $D=4$, we give anestimate on the transition temperature $T_d$ as the ratio $T_d/M$ to the mass$M$ which has been measured on the lattice at zero temperature and iscalculable also at finite temperature. We show that the order of the phasetransition at $T_d$ is the second order for $SU(2)$ and weakly first order for$SU(3)$ Yang-Mills theory. We elucidate what is the mechanism for quarkconfinement and deconfinement at finite temperature and why the phasetransition occurs at a certain temperature. These initial results are obtainedeasily based on the analytical calculations of the "one-loop type" in the firstapproximation. We discuss also how these results are improved to eliminate theartifacts obtained for some thermodynamic observables.

Confinement/deconfinement transition temperature from the Polyakov loop potential and gauge-invariant gluon mass