Remarks on the Gribov Problem in Direct Maximal Center Gauge

TL;DR

This paper investigates the Gribov problem in direct maximal center gauge, comparing different gauge fixing methods and their impact on vortex fitting and string tension measurements in lattice gauge theory.

Contribution

It provides a comparative analysis of gauge fixing techniques and their effects on vortex fitting quality and string tension estimates in lattice gauge theory.

Findings

Simulated annealing achieves the best fit when all links are included.

Over-relaxation yields the best fit in the exterior volume outside P-plaquettes.

String tension estimates vary significantly depending on the fitting criteria.

Abstract

We review the equivalence of maximal center gauge fixing to the problem of finding the best fit, to a given lattice gauge field, by a thin vortex configuration. This fit is necessarily worst at the location of P-plaquettes. We then compare the fits achieved in Gribov copies generated by (i) over-relaxation; (ii) over-relaxation after Landau gauge preconditioning; and (iii) simulated annealing. Simulated annealing yields the best fit if all links on the lattice are included, but the situation changes if we consider only the lattice volume exterior to P-plaquettes. In this exterior region, the fit is best for Gribov copies generated by over-relaxation, and worst for Gribov copies generated after Landau gauge preconditioning. The two fitting criteria (including or not including the P-plaquettes) yield string tensions differing by -34% to +20% respectively, relative to the full string tension. Our usual procedure (``quenched minimization'') seems to be a compromise between these criteria, and yields string tensions at an intermediate value close to the full string tension.