Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions

Abstract

The covariant gauges are known to suffer from the Gribov problem: even after fixing a gauge non-perturbatively, there may still exist residual copies which are physically equivalent to each other, called Gribov copies. While the influence of Gribov copies in the relevant quantities such as gluon propagators has been heavily debated in recent studies, the significance of the role they play in the Faddeev--Popov procedure is hardly doubted. We concentrate on Gribov copies in the first Gribov region, i.e., the space of Gribov copies at which the Faddeev--Popov operator is strictly positive (semi)definite. We investigate compact U($1$) as the prototypical model of the more complicated standard model group SU($N_c$). With our Graphical Processing Unit (GPU) implementation of the relaxation method we collect up to a few million Gribov copies per orbit. We show that the numbers of Gribov copies even in the first Gribov region increase exponentially in two, three and four dimensions. Furthermore, we provide strong indication that the number of Gribov copies is gauge orbit dependent.

8 pages, 7 figures. V2: version accepted at PRD

References in corpus (7)

• Phase transitions and configuration space topology
• Enumerating Gribov copies on the lattice
• Decontracted double BRST on the lattice
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Exploring the energy landscape of XY models
• Phase with no mass gap in non-perturbatively gauge-fixed Yang--Mills theory
• Gribov copies and anomalous scaling

Cited by in corpus (6)

• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Gribov horizon and Gribov copies effect in lattice Coulomb gauge
• Gauge-fixing on the Lattice via Orbifolding
• A window on infrared QCD with small expansion parameters
• Gribov copies in the quark propagator