Gribov copies in the minimal Landau gauge: The influence on gluon and ghost propagators

Abstract

We study the influence of Gribov copies on gluon and ghost propagators, evaluated numerically in pure SU (2) lattice gauge theory in the minimal Landau gauge. Simulations are done at four different values of β (namely β = 0, 0.8, 1.6 and 2.7 ) and for volumes up to 16 4 (up to 24 4 at β = 1.6). For the gluon propagator, Gribov noise seems to be of the order of magnitude of the numerical accuracy, even at very small values of the coupling β. On the contrary, for the ghost propagator, Gribov noise is clearly observable for the three values of β in the strong-coupling regime. In particular, data corresponding to the minimal Landau gauge are always smaller than those obtained in a generic Landau gauge. This result can be qualitatively explained.Gauge theories, being invariant under local gauge transformations, are systems with redundant dynamical variables, which do not represent true dynamical degrees of freedom. The objects of interest are not the gauge fields themselves, but rather the classes (orbits) of gauge-related fields. The elimination of such redundant gauge degrees of freedom is essential for understanding and extracting physical information from these theories. This is usually done by a method called gauge fixing, which is based on the assumption that a gauge-fixing condition can be found which uniquely determines a representative gauge field on each orbit. It was pointed out by Gribov [1] that the standard gauge-fixing conditions used for perturbative calculations do not in fact fix the gauge fields uniquely: for a non-abelian gauge theory, in the Coulomb or in the Landau gauge, there are many gauge equivalent configurations satisfying the Coulomb or Landau transversality condition. The existence of these Gribov copies does not affect the results from perturbation theory, but their elimination could play a crucial role for non-perturbative features of these theories.One of the celebrated advantages of lattice gauge theories is that the lattice provides a regularization which makes the gauge group compact, so that the Gibbs average of any gauge-invariant quantity is well-defined and therefore gauge fixing is, in principle, not required. However, because

Numerical study of the fundamental modular region in the minimal Landau gauge

“…This is in agreement with the result obtained in reference [11], in which Gribov noise has been observed for the ghost propagator. In all cases, the effect is small but clearly detectable for the values of β in the strong-coupling region.…”

“…As for the volume dependence, it seems that the horizon tensor becomes closer and closer 11 We have found only in very few cases a value of r larger than 8/3, which indicates the existence of other stationary points near the minimum at τ = 0 (see Figure 2). 12 See data for the average of the non-diagonal elements in Table 3.…”

“…To date, relatively few studies [10,11,16] have been done in order to analyze the influence of Gribov copies (Gribov noise) on some lattice quantities. However, these numerical studies have never tried to characterize the "geometry" of the orbit space.…”

QCD Green functions and their application to hadron physics

“…A lot of effort has been devoted to an estimate of the corresponding effects in lattice studies, see e.g. [15][16][17][18][19][20][21]. These provided evidence that for Green functions of lattice gauge theory the effects due to Gribov copies are not too large.…”

Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function

“…In this region, the perturbative running coupling s ð 2 Þ diverges, and the confinement effect is extremely large, so that quark-gluon degrees of freedom are hidden and the system is described by hadrons. So far, the gluon propagator has been studied mainly in the Landau gauge both in analytic framework [14,15,[17][18][19] and in lattice QCD [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In the UV region, the gluon propagator takes a massless perturbative form, 1=p 2 , apart from the tensor factor.…”