A Non-Abelian Variation on the Savvidy Vacuum of the Yang-Mills Gauge Theory

TL;DR

This paper calculates a one-loop effective potential for a non-Abelian gauge configuration using the background field method, proposing a new background field that avoids certain singularities and technical issues of previous approaches, aligning with Savvidy's results.

Contribution

It introduces a novel non-Abelian background field configuration that circumvents coordinate singularities and lattice technical issues, advancing the understanding of the Savvidy vacuum in Yang-Mills theory.

Findings

Avoids coordinate singularity Det B_i^a=0

Generates constant color magnetic field via commutator terms

Qualitatively reproduces Savvidy's results

Abstract

As a prelude to a truly non-perturbative evaluation of the effective potential in terms of lattice QCD, the one loop effective potential for a non-Abelian gauge configuration is calculated using the background field method. Through a non-trivial correlation between the space and color orientations the new background field avoids the possible coordinate singularity, ${\rm Det}B_i^a=0$, observed recently by Ken Johnson and his collaborators in their Schr\"{o}dinger functional study of the SU(2) Yang-Mills theory. In addition, since our ansatz generates a constant color magnetic field through the commutator terms rather than derivative terms, many of the technical drawbacks the Savvidy ansatz suffers on a lattice can be avoided. Our one loop study yields qualitatively the same result as that of Savvidy's.