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The Gribov problem in the presence of a background field is analyzed: inparticular, we study the Gribov copies equation in the Landau-De Witt gauge aswell as the semi-classical Gribov gap equation. As background field, we choosethe simplest non-trivial one which corresponds to a constant gauge potentialwith non-vanishing component along the Euclidean time direction. This kind ofconstant non-Abelian background fields is very relevant in relation with (thecomputation of) the Polyakov loop but it also appears when one considers thenon-Abelian Schwinger effect. We show that the Gribov copies equation isaffected directly by the presence of the background field, constructing anexplicit example. The analysis of the Gribov gap equation shows that the largerthe background field, the smaller the Gribov mass parameter. These resultsstrongly suggest that the relevance of the Gribov copies (from the pathintegral point of view) decreases as the size of the background fieldincreases.

The Gribov problem in presence of background field for $SU(2)$ Yang-Mills theory