The one loop MSbar static potential in the Gribov-Zwanziger Lagrangian

TL;DR

This paper calculates the one-loop static potential within the Gribov-Zwanziger framework in the MSbar scheme, revealing modifications due to the Gribov mass and analyzing the potential's behavior at different distances.

Contribution

It provides the explicit one-loop form of the static potential in the Gribov-Zwanziger Lagrangian and derives the gap equation for the Gribov mass in the MOM scheme.

Findings

The static potential reduces to the usual perturbative form as gamma approaches zero.

A dipole behavior appears at short distances, but a linearly rising potential does not emerge at large distances.

Enhancement of the Zwanziger ghost propagator is demonstrated when the gap equation is satisfied.

Abstract

We compute the static potential in the Gribov-Zwanziger Lagrangian as a function of the Gribov mass, gamma, in the MSbar scheme in the Landau gauge at one loop. The usual gauge independent one loop perturbative static potential is recovered in the limit as gamma -> 0. By contrast the Gribov-Zwanziger static potential contains the term gamma^2/(p^2)^2. However, the linearly rising potential in coordinate space as a function of the radial variable r does not emerge due to a compensating behaviour as r -> infty. Though in the short distance limit a dipole behaviour is present. We also demonstrate enhancement in the propagator of the bosonic localizing Zwanziger ghost field when the one loop Gribov gap equation is satisfied. The explicit form of the one loop gap equation for the Gribov mass parameter is also computed in the MOM scheme and the zero momentum value of the renormalization group invariant effective coupling constant is shown to be the same value as that in the MSbar scheme.