More on the renormalization of the horizon function of the Gribov-Zwanziger action and the Kugo-Ojima Green function(s)

TL;DR

This paper clarifies the unique form of the horizon function in the Gribov-Zwanziger framework, emphasizing its role in ensuring renormalizability and discussing the proper definition of Kugo-Ojima functions after renormalization.

Contribution

It demonstrates the uniqueness of the horizon function based on renormalizability requirements and clarifies the conditions for defining Kugo-Ojima functions.

Findings

Only one horizon function is consistent with renormalizability.

Relations from other horizon functions lack a proper quantum field theory interpretation.

Kugo-Ojima functions are well-defined only after renormalization.

Abstract

In this paper we provide strong evidence that there is no ambiguity in the choice of the horizon function underlying the Gribov-Zwanziger action. We show that there is only one correct possibility which is determined by the requirement of multiplicative renormalizability. As a consequence, this means that relations derived from other horizon functions cannot be given a consistent interpretation in terms of a local and renormalizable quantum field theory. In addition, we also discuss that the Kugo-Ojima functions $u(p^2)$ and $w(p^2)$ can only be defined after renormalization of the underlying Green function(s).