Gribov horizon and Gribov copies effect in lattice Coulomb gauge

Abstract

Following a recent proposal by Cooper and Zwanziger [1] 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 FaddeevPopov 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 VC 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.

Hamiltonian Approach to QCD in Coulomb Gauge: A Survey of Recent Results

“…, 10000 of times, starting each time from a different random gauge transformation of the original configuration. In general, the number 푟 of gaugefixing trials is indicative of the number of Gribov copies included, even though the exact relation is complicated and nonlinear [40].) Although we did not find a strict saturation for a sufficiently large number of gauge-fixing attempts, the IR exponent of the ghost form factor is compatible with the continuum result of ≃ 1 (see Figure 9), in agreement with the sum rule (16).…”

“…As argued in [39], the "lowest copy" method should yield results closer to the continuum theory. We have used the "lowest copy" (lc) method to recalculate the ghost and gluon propagator; see [40]. While the gluon propagator is basically the same as obtained with the "best copy" (bc) method (see Figure 7), the ghost form factor gets further enhanced in the IR as the number of gauge-fixing attempts increases (see Figure 8).…”

A non-perturbative study of matter field propagators in Euclidean Yang–Mills theory in linear covariant, Curci–Ferrari and maximal Abelian gauges

“…There are two notable examples of gauge choices which also possess a Hermitean Faddeev-Popov operator: the maximal Abelian and Coulomb gauges. For such gauges, an explicit construction of the Gribov-Zwanziger action and its refinement was performed, see [48][49][50][51][52][53][54][55][56][57][58][59][60][61]. In spite of this fact, these gauges have their own peculiarities and the development of the refined Gribov-Zwanziger scenario for them is not at the same level as in the Landau gauge.…”

Chiral symmetry restoration at finite temperature within the Hamiltonian approach to QCD in Coulomb gauge

“…The Coulomb string tension σ C entering this expression sets the overall scale in the present model. Lattice and continuum calculations [24][25][26][27][28] For a numerical evaluation, eq. ( 24) is not directly useful, since the entire calculation is dominated by the pole of the Coulomb potential for the single frequency Ω n = Ω l which -in constrast to the T = 0 equation discussed below -is not lifted by the integration measure.…”

“…The Coulomb string tension σ C entering this expression sets the overall scale in the present model. Lattice and continuum calculations [24][25][26][27][28] favour values σ C /σ ≈ 2 . .…”