An all-order proof of the equivalence between Gribov's no-pole and Zwanziger's horizon conditions

TL;DR

This paper proves, to all orders, that Gribov's no-pole condition and Zwanziger's horizon condition are equivalent in the quantization of non-Abelian gauge theories, clarifying their relationship in the Gribov-Zwanziger framework.

Contribution

It provides an exact, all-order proof of the equivalence between Gribov's no-pole condition and Zwanziger's horizon condition in SU(N) Yang-Mills theory.

Findings

Established the all-order equivalence between the two conditions.

Derived a compact expression for the ghost propagator in terms of gauge fields.

Connected the no-pole condition to the expectation value of Zwanziger's horizon function.

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 Gribov's no-pole condition, can be implemented by demanding a nonvanishing expectation value for a functional of the gauge fields that turns out to be Zwanziger's horizon function. The action allowing to implement this condition is the Gribov-Zwanziger action. This establishes in a precise way the equivalence between Gribov's no-pole condition and Zwanziger's horizon condition.