Ghost sector and geometry in minimal Landau gauge: further constraining the infinite-volume limit

TL;DR

This paper derives bounds for the ghost propagator in Yang-Mills theories using eigenvalues of the Faddeev-Popov matrix, verified through lattice simulations, and offers insights into the infinite-volume limit and Gribov effects.

Contribution

It provides new bounds for the ghost propagator based on eigenvalues and relates the infinite-volume limit to the geometry of the Gribov region in lattice gauge theory.

Findings

Bounds for the ghost propagator are verified with lattice data.

The approach explains the absence of ghost enhancement in the infrared.

A relation between eigenvalues, geometry, and the Gribov horizon is established.

Abstract

We present improved upper and lower bounds for the momentum-space ghost propagator of Yang-Mills theories in terms of the two smallest nonzero eigenvalues (and their corresponding eigenvectors) of the Faddeev-Popov matrix. These results are verified using data from four-dimensional numerical simulations of SU(2) lattice gauge theory in minimal Landau gauge at beta = 2.2, for lattice sides N = 16, 32, 48 and 64. Gribov-copy effects are discussed by considering four different sets of numerical minima. We then present a lower bound for the smallest nonzero eigenvalue of the Faddeev-Popov matrix in terms of the smallest nonzero momentum on the lattice and of a parameter characterizing the geometry of the first Gribov region $\Omega$. This allows a simple and intuitive description of the infinite-volume limit in the ghost sector. In particular, we show how nonperturbative effects may be quantified by the rate at which typical thermalized and gauge-fixed configurations approach the boundary of Omega, known as the first Gribov horizon. As a result, a simple and concrete explanation emerges for why lattice studies do not observe an enhanced ghost propagator in the deep infrared limit. Most of the simulations have been performed on the Blue Gene/P--IBM supercomputer shared by Rice University and S\~ao Paulo University.