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

TL;DR

This paper analytically derives the confinement/deconfinement transition temperature in SU(N) Yang-Mills theory using a gauge-invariant gluonic mass and the Polyakov loop potential, clarifying the phase transition mechanism.

Contribution

It introduces a novel reformulation of Yang-Mills theory with a gauge-invariant gluonic mass to analytically determine the phase transition and transition temperature.

Findings

Transition is second order for SU(2) and weakly first order for SU(3).

Estimated transition temperature ratio T_d/M matches lattice measurements.

Analytical approach provides insights into confinement and deconfinement mechanisms.

Abstract

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 $D>2$. 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 $M$. For $D=4$, we give an estimate 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 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. We elucidate what is the mechanism for quark confinement and deconfinement at finite temperature and why the phase transition occurs at a certain temperature. These initial results are obtained easily based on the analytical calculations of the "one-loop type" in the first approximation. We discuss also how these results are improved to eliminate the artifacts obtained for some thermodynamic observables.