Gribov horizon and Gribov copies effect in lattice Coulomb gauge

TL;DR

This study uses lattice simulations to examine how selecting Gribov copies near the horizon affects Coulomb gauge propagators and confinement, revealing that such choices can overshoot continuum predictions and that further restrictions are needed for physical relevance.

Contribution

It demonstrates that choosing Gribov copies with minimal eigenvalues influences Coulomb gauge propagators and highlights the necessity for additional gauge restrictions to ensure physical meaningfulness.

Findings

Selecting Gribov copies near the horizon affects ghost propagators.

The Coulomb potential from Faddeev-Popov eigenvalues becomes physically meaningless.

Alternative Coulomb potential definitions are less affected by small eigenvalues.

Abstract

Following a recent proposal by Cooper and Zwanziger we investigate via $SU(2)$ lattice simulations the effect on the Coulomb gauge propagators and on the Gribov-Zwanziger confinement mechanism of selecting the Gribov copy with the smallest non-trivial eigenvalue of the Faddeev-Popov operator, i.e.~the one closest to the Gribov horizon. Although such choice of gauge drives the ghost propagator towards the prediction of continuum calculations, we find that it actually overshoots the goal. With increasing computer time, we observe that Gribov copies with arbitrarily small eigenvalues can be found. For such a method to work one would therefore need further restrictions on the gauge condition to isolate the physically relevant copies, since e.g.~the Coulomb potential $V_C$ defined through the Faddeev-Popov operator becomes otherwise physically meaningless. Interestingly, the Coulomb potential alternatively defined through temporal link correlators is only marginally affected by the smallness of the eigenvalues.