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The properties of the Gribov region in SU(2) Euclidean Yang-Mills theories inthe maximal Abelian gauge are investigated. This region turns out to be boundedin 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 domainof integration in the path integral to the Gribov region is scrutinized. Owingto the unboundedness in the diagonal direction, the invariance with respect toAbelian transformations is preserved, a property which is at the origin of thelocal U(1) Ward identity of the maximal Abelian gauge.

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