Abstract
The Gribov problem in the presence of a background field is analyzed: in particular, we study the Gribov copies equation in the Landau-De Witt gauge as well as the semi-classical Gribov gap equation. As background field, we choose the simplest non-trivial one which corresponds to a constant gauge potential with non-vanishing component along the Euclidean time direction. This kind of constant non-Abelian background fields is very relevant in relation with (the computation of) the Polyakov loop but it also appears when one considers the non-Abelian Schwinger effect. We show that the Gribov copies equation is affected directly by the presence of the background field, constructing an explicit example. The analysis of the Gribov gap equation shows that the larger the background field, the smaller the Gribov mass parameter. These results strongly suggest that the relevance of the Gribov copies (from the path integral point of view) decreases as the size of the background field increases.
Highlights
The main tool to compute observable quantities in QFT is perturbation theory
Explicit examples have been constructed in which the norm of the Gribov copies satisfying the usual boundary conditions increases when the size of the background field is very large
We have shown that the larger is the size of the background gauge potential, the smaller is the corresponding Gribov mass
Summary
The main tool to compute observable quantities in QFT is perturbation theory. In gauge theories, and in Yang–Mills (YM) theory in particular, a fundamental problem to solve in order to compute physical quantities is the over-counting of degrees of freedom related to gauge invariance (for a detailed analysis see [1]). When the space– time geometry is flat and the topology trivial, this method is able to reproduce the usual perturbation theory encoding, at the same time, the effects related to the elimination of the Gribov copies It allows the computation of the glueball masses in excellent agreement with the lattice data [22,23,24]. One of the most relevant applications of the background field method is the computation of the (vacuum expectation value of the) Polyakov loop [32] in which the presence of the Polyakov loop manifests itself as a constant background field with component along the Euclidean time.4 Another very important non-perturbative phenomenon in which the presence of a background gauge field plays a key role is the (both Abelian and non-Abelian) Schwinger effect [36,37,38].
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