An all-order proof of the equivalence between Gribovʼs no-pole and Zwanzigerʼs horizon conditions

M.A.L Capri,D Dudal,M.S Guimaraes,L.F Palhares,S.P Sorella

doi:10.1016/j.physletb.2013.01.039

M.A.L Capri, D Dudal + Show 3 more

Open Access

https://doi.org/10.1016/j.physletb.2013.01.039

Copy DOI

Abstract

The quantization of non-Abelian gauge theories is known to be plagued by Gribov copies. Typical examples are the copies related to zero modes of the Faddeev–Popov operator, which give rise to singularities in the ghost propagator. In this work we present an exact and compact expression for the ghost propagator as a function of external gauge fields, in SU(N) Yang–Mills theory in the Landau gauge. It is shown, to all orders, that the condition for the ghost propagator not to have a pole, the so-called Gribovʼs no-pole condition, can be implemented by demanding a non-vanishing expectation value for a functional of the gauge fields that turns out to be Zwanzigerʼs horizon function. The action allowing to implement this condition is the Gribov–Zwanziger action. This establishes in a precise way the equivalence between Gribovʼs no-pole condition and Zwanzigerʼs horizon condition.

Highlights

It is well known that the gauge fixing quantization procedure of Yang-Mills gauge theories suffers from ambiguities related to the existence of the so called Gribov copies [1]
In this work we present an exact and compact expression for the ghost propagator as a function of external gauge fields, in SU(N) Yang-Mills theory in the Landau gauge
In the Landau gauge it consists of restricting the domain of integration in the Euclidean functional integral to the Gribov region Ω, defined as the region in field space where the Faddeev-Popov operator is strictly positive

Summary

Introduction

It is well known that the gauge fixing quantization procedure of Yang-Mills gauge theories suffers from ambiguities related to the existence of the so called Gribov copies [1]. Following [1], The implementation of the restriction to Ω amounts to impose that the ghost propagator, i.e. the inverse of the Faddeev-Popov operator, has no poles at finite non-vanishing values of the momentum k This implies that, within the region Ω, the ghost propagator remains always positive, namely the Gribov horizon ∂Ω is never crossed. In [8], it was shown that Zwanziger’s horizon function can be matched to the ghost form factor, defined through the no-pole condition, up to the third order in the external gauge field. This condition cannot be realized within the Faddeev-Popov functional measure, as pointed out in [9] Instead, it can be consistently implemented by employing the Gribov-Zwanziger action, and corresponds to the Zwanziger’s horizon condition.

Gribov’s no-pole condition

Zwanziger’s horizon condition

The exact ghost form factor and the horizon function

Conclusion