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

Abstract

The recently proposed algebraic approach is used for calculating the heat kernel on covariantly constant background to study the one-loop effective action in the non-Abelian gauge theory. The general case of arbitrary space-time dimension, arbitrary compact simple gauge group and arbitrary matter is considered and a covariantly constant gauge field strength of the most general form, which has many independent color and space-time invariants, and covariantly constant scalar fields as a background (Savvidy type chromomagnetic vacuum) is assumed. The explicit formulas for all the needed heat kernels and zeta-functions are obtained. A new method is proposed to study the vacuum stability and it will be shown that the background field configurations with covariantly constant chromomagnetic fields can be stable only in the case when the number of independent field invariants is greater than one and the values of these invariants differ little 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 of dimension greater than four.