Study of the properties of the Gribov region in SU(N) Euclidean Yang-Mills theories in the maximal Abelian gauge

In this paper we address the issue of the Gribov copies in SU(N), N > 2, Euclidean Yang-Mills theories quantized in the maximal Abelian gauge. A few properties of the Gribov region in this gauge are established. Similar to the case of SU(2), the Gribov region turns out to be convex, bounded along the off-diagonal directions in field space, and unbounded along the diagonal ones. The implementation of the restriction to the Gribov region in the functional integral is discussed through the introduction of the horizon function, whose construction will be outlined in detail. The influence of this restriction on the behavior of the gluon and ghost propagators of the theory is also investigated together with a set of dimension 2 condensates.