Phase structure of $SU(2)$ Yang-Mills theory with global center symmetry

Abstract

Retaining only the `timelike' component $A_0$ of the vector potential a skelet model with explicit global center symmetry is constructed for $SU(2)$ Yang-Mills theory. It is shown that the $A_0$ gluon vacuum is equivalent with the 4-dimensional Coulomb-gas. In 1--loop approximation, the effective theory of the skelet model exhibits non-local self-interaction. A coupling constant $\lambda$ is found that governs the loop expansion. For $\lambda < 1$ the effective theory does not confine (the string tension vanishes), whereas for $\lambda = 1$ confinement (with non-vanishing string tension) takes place. In the deconfined phase the $A_0$ gluons exhibit non-zero rest mass and global center symmetry is broken by the vacuum state. Confinement sets on for $\lambda =1$ when the rest mass of $A_0$ gluons vanishes and the global center symmetry of the vacuum is restored.