# Invariant measure in hot gauge theories

## TL;DR

This paper studies the invariant measure of the $A_0$ gauge field in finite temperature gauge theories, revealing cancellations and the role of Z(N) symmetry in confinement and $A_0$ condensation.

## Contribution

It demonstrates the cancellation of the naive measure at all temperatures and clarifies the role of the invariant measure and Z(N) symmetry in gauge theories.

## Key findings

- Naive measure cancels at all temperatures.
- No cancellation for Z(N) symmetric distribution.
- Z(N) symmetry breaking driven by gluonic potential energy.

## Abstract

We investigate properties of the invariant measure for the $A_0$ gauge field in finite temperature gauge theories both on the lattice and in the continuum theory. We have found the cancellation of the naive measure in both cases. The result is quite general and holds at any finite temperature. We demonstrate, however, that there is no cancellation at any temperature for the invariant measure contribution understood as Z(N) symmetrical distribution of gauge field configurations. The spontaneous breakdown of Z(N) global symmetry is entirely due to the potential energy term of the gluonic interaction in the effective potential. The effects of this measure on the effective action, mechanism of confinement and $A_0$ condensation are discussed.