Study of the Gribov region in Euclidean Yang-Mills theories in the maximal Abelian gauge

TL;DR

This paper investigates the properties of the Gribov region in SU(2) Euclidean Yang-Mills theories within the maximal Abelian gauge, revealing its boundedness off-diagonal and unbounded diagonal directions, and analyzing the implications for BRST invariance and Abelian symmetry.

Contribution

It provides a detailed analysis of the Gribov region's geometric properties and their impact on symmetry invariances in the maximal Abelian gauge.

Findings

The Gribov region is bounded off-diagonal and unbounded along the diagonal.

The unboundedness preserves Abelian invariance and the U(1) Ward identity.

The study clarifies the role of the Gribov restriction in gauge symmetry.

Abstract

The properties of the Gribov region in SU(2) Euclidean Yang-Mills theories in the maximal Abelian gauge are investigated. This region turns out to be bounded in all off-diagonal directions, while it is unbounded along the diagonal one. The soft breaking of the BRST invariance due to the restriction of the domain of integration in the path integral to the Gribov region is scrutinized. Owing to the unboundedness in the diagonal direction, the invariance with respect to Abelian transformations is preserved, a property which is at the origin of the local U(1) Ward identity of the maximal Abelian gauge.