Field Strength Formulation of SU(2) Yang-Mills Theory in the Maximal Abelian Gauge: Perturbation Theory

TL;DR

This paper reformulates SU(2) Yang-Mills theory in the maximal Abelian gauge by integrating out non-Abelian components, introducing a tensor field, and demonstrating that semiclassical expansion reproduces standard perturbation theory results.

Contribution

It introduces a new formulation of SU(2) Yang-Mills theory using an Abelian tensor field, connecting semiclassical expansion to conventional perturbation theory.

Findings

Reproduces the one-loop β-function for the running coupling constant.

Shows semiclassical expansion around trivial solution matches standard perturbation theory.

Derives saddle point equations related to non-perturbative interactions.

Abstract

We present a reformulation of SU(2) Yang-Mills theory in the maximal Abelian gauge, where the non-Abelian gauge field components are exactly integrated out at the expense of a new Abelian tensor field. The latter can be treated in a semiclassical approximation and the corresponding saddle point equation is derived. Besides the non-trivial solutions, which are presumably related to non-perturbative interactions for the Abelian gauge field, the equation of motion for the tensor fields allows for a trivial solution as well. We show that the semiclassical expansion around this trivial solution is equivalent to the standard perturbation theory. In particular, we calculate the one-loop $\beta$-function for the running coupling constant in this approach and reproduce the standard result.