# Gribov Copies in the Maximally Abelian Gauge and Confinement

## TL;DR

This study investigates how Gribov copies affect the extraction of $U(1)$ fields, monopole distributions, and string tensions in the Maximally Abelian gauge, finding that the resulting ambiguities are relatively limited for confinement studies.

## Contribution

It provides a detailed analysis of Gribov copy effects on monopole and $U(1)$ field measurements, showing limited impact on string tension estimates in the Maximally Abelian gauge.

## Key findings

- String tension differences from Gribov copies are about 20%.
- Monopole distributions vary significantly between copies.
- $U(1)$ fluxes are highly correlated across Gribov copies.

## Abstract

We fix $SU(2)$ lattice gauge fields to the Maximally Abelian gauge in both three and four dimensions. We extract the corresponding $U(1)$ fields and monopole current densities and calculate separately the confining string tensions arising from these $U(1)$ fields and monopole `condensates'. We generate multiple Gribov copies and study how the $U(1)$ fields and monopole distributions vary between these different copies. As expected, we find substantial variations in the number of monopoles, their locations and in the values of the $U(1)$ field strengths. The string tensions extracted from `extreme' Gribov copies also differ but this difference appears to be no more than about 20\%. We also directly compare the fields of different Gribov copies. We find that on the distance scales relevant to confinement the $U(1)$ and monopole fluxes that disorder Wilson loops are highly correlated between these different Gribov copies. All this suggests that while there is indeed a Gribov copy problem the resulting ambiguity is, in this gauge and for the study of confinement, of limited importance.