The Gribov parameter and the dimension two gluon condensate in Euclidean Yang-Mills theories in the Landau gauge

TL;DR

This paper proves the all-order renormalizability of a modified Yang-Mills action incorporating the Gribov restriction and the A^2 condensate, evaluates the effective action at one-loop, and explores implications for gluon and ghost propagators.

Contribution

It introduces a renormalizable framework combining the Gribov restriction with the A^2 condensate and analyzes vacuum properties and propagator behaviors.

Findings

Explicit values for the Gribov and mass parameters are obtained.

Vacuum energy remains positive in the original formulation, with scheme-dependent results when including <A^2>.

Infrared suppression of gluons and enhancement of ghosts are confirmed, aligning with other nonperturbative studies.

Abstract

The local composite operator A^2 is added to the Zwanziger action, which implements the restriction to the Gribov region in Euclidean Yang-Mills theories in the Landau gauge. We prove the renormalizability of this action to all orders of perturbation theory. This allows to study the dimension two gluon condensate <A^2> by the local composite operator formalism when the restriction is taken into account. The effective action is evaluated at one-loop order in the MSbar scheme. We obtain explicit values for the Gribov parameter and for the mass parameter due to <A^2>, but the expansion parameter turns out to be rather large. Furthermore, an optimization of the perturbative expansion in order to reduce the dependence on the renormalization scheme is performed. The properties of the vacuum energy, with or without <A^2>, are investigated. It is shown that in the original Gribov-Zwanziger formulation (without <A^2>), the vacuum energy is always positive at 1-loop order, independently from the renormalization scheme and scale. With <A^2>, we are unable to come to a definite conclusion at the order considered. In the MSbar scheme, we still find a positive vacuum energy, again with a relatively large expansion parameter, but there are renormalization schemes in which the vacuum energy is negative, albeit the dependence on the scheme itself appears to be strong. We recover the well known consequences of the restriction, and this in the presence of <A^2>: an infrared suppression of the gluon propagator and an enhancement of the ghost propagator. This behaviour is in qualitative agreement with the results obtained from the studies of the Schwinger-Dyson equations and from lattice simulations.