Gribov's horizon and the ghost dressing function

TL;DR

This paper investigates the relationship between the horizon function and ghost dressing functions in gauge theories, analyzing the implications of the horizon gap equation and lattice results for the behavior of the ghost dressing function at zero momentum.

Contribution

It clarifies the impact of the horizon gap equation on the ghost dressing function and discusses the non-multiplicative renormalizability of related functions.

Findings

The solution w(0)=0 is not acceptable under the horizon gap equation.

The renormalised ghost dressing function is finite and non-zero at zero momentum.

Lattice data suggests F_R(0) ≈ 2.2 at 1.5 GeV.

Abstract

We study a relation recently derived by K. Kondo at zero momentum between the Zwanziger's horizon function, the ghost dressing function and Kugo's functions $u$ and $w$. We agree with this result as far as bare quantities are considered. However, assuming the validity of the horizon gap equation, we argue that the solution $w(0)=0$ is not acceptable since it would lead to a vanishing renormalised ghost dressing function. On the contrary, when the cut-off goes to infinity, $u(0) \to \infty$, $w(0) \to -\infty$ such that $u(0)+w(0) \to -1$. Furthermore $w$ and $u$ are not multiplicatively renormalisable. Relaxing the gap equation allows $w(0)=0$ with $u(0) \to -1$. In both cases the bare ghost dressing function, $F(0,\Lambda)$, goes logarithmically to infinity at infinite cut-off. We show that, although the lattice results provide bare results not so different from the $F(0,\Lambda)=3$ solution, this is an accident due to the fact that the lattice cut-offs lie in the range 1-3 GeV$^{-1}$. We show that the renormalised ghost dressing function should be finite and non-zero at zero momentum and can be reliably estimated on the lattice up to powers of the lattice spacing ; from published data on a $80^4$ lattice at $\beta=5.7$ we obtain $F_R(0,\mu=1.5$ GeV)$\simeq 2.2$.