Covariant algebraic calculation of the one-loop effective potential in non-Abelian gauge theory and a new approach to stability problem

TL;DR

This paper develops an algebraic method to calculate the one-loop effective potential in non-Abelian gauge theories with covariantly constant backgrounds, revealing stability conditions dependent on space-time dimensions and invariants.

Contribution

It introduces a new algebraic approach for heat kernel calculations and proposes a novel stability analysis method for gauge field configurations.

Findings

Stable vacua require multiple independent invariants with similar values.

Stability is only possible in space-times with dimensions five or higher.

Explicit formulas for heat kernels and zeta-functions are derived.

Abstract

We use our recently proposed algebraic approach for calculating the heat kernel associated with the Laplace operator to calculate the one-loop effective action in the non-Abelian gauge theory. We consider the most general case of arbitrary space-time dimension, arbitrary compact simple gauge group and arbitrary matter and assume a covariantly constant gauge field strength of the most general form, having many independent color and space-time invariants (Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields as a background. The explicit formulas for all the needed heat kernels and zeta-functions are obtained. We propose a new method to study the vacuum stability and show that the background field configurations with covariantly constant chromomagnetic fields can be stable only in the case when more than one independent field invariants are present and the values of these invariants differ not greatly from each other. The role of space-time dimension is analyzed in this connection and it is shown that this is possible only in space-times with dimensions not less than five $d\geq 5$.