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 Becchi-Rouet-Stora-Tyutin 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.

• Received 18 November 2008

DOI:https://doi.org/10.1103/PhysRevD.79.025019