Living with spontaneously broken BRST symmetry. I. Physical states and cohomology

Abstract

Abstract:We address the issue of Becchi-Rouet-Stora-Tyutin (BRST) symmetry breaking in the Gribov-Zwanziger (GZ) model, a local, renormalizable, nonperturbative approach to QCD. Explicit calculation of several examples reveals that BRST symmetry breaking apparently afflicts the unphysical sector of the theory, but may be unbroken where needed, in cases of physical interest. Specifically, the BRSTexact part of the conserved energy-momentum tensor and the BRST-exact term in the Kugo-Ojima confinement condition both have vanishing expectation value. We analyze the origin of the breaking of BRST symmetry in the GZ model, and obtain a useful sufficient condition that determines which operators preserve BRST.Observables of the GZ theory are required to be invariant under a certain group of symmetries that includes not only BRST but also others. The definition of observables is thereby sharpened, and excludes all operators known to us that break BRST invariance. We take as a hypothesis that BRST symmetry is unbroken by this class of observables. If the hypothesis holds, BRST breaking is relegated to the unphysical sector of the GZ theory, and its physical states are obtained by the usual cohomological BRST construction. The fact that the horizon condition and the Kugo-Ojima confinement criterion coincide assures that color is confined in the GZ theory.

Constraining the gauge-fixed Lagrangian in minimal Landau gauge

“…While it is possible to get decent agreement with suitable manipulations, see e. g. [3,35,37,52] for reviews, or suitable adjusted expansion points in semi-perturbative methods, see [4] for a review, a difference in the DSEs would explain the observed problems. It has also been variously argued that additional gauge-fixing terms may be necessary to replicate the lattice minimal Landau gauge [4,20,37,[53][54][55][56][57][58], including various limiting procedures from all Gribov regions [17,19,59] or even from the full gauge orbit [18,23,24,60]. Such ideas have been variously implemented approximately in calculations [16,17,19,25,59], yielding acceptable agreement with lattice results.…”

Once again on the Gribov horizon

“…Last but not least remark. Quite recently a reformulation of the original GZ theory has been proposed [22] to improve quantum properties of the theory concerning the BRST symmetry and its nilpotency. In this connection it should be definitely noted that the reformulation was performed with the help of change of variables which violates the transformation laws of fields under the Poincare group.…”

“…In this connection it should be definitely noted that the reformulation was performed with the help of change of variables which violates the transformation laws of fields under the Poincare group. If we will consider the GZ theory [1,2] formulated in accordance with general principles of quantum field theory then we should consider the new theory [22] as ill-defined and vice versa. These theories can not exist simultaneously as equivalent.…”

Gribov horizon, Polyakov loop and finite temperature

“…As such, it is a soft breaking and the ultraviolet divergences can be controlled at the quantum level. The properties of the soft breaking of the BRST symmetry of the GZ theory and its relation with confinement have been object of intensive investigation in recent years, see [38,39,40,41,42,43,44,45,46,47]. Here, it suffices to mention that the broken identity (2.16) is connected with the restriction to the Gribov region Ω.…”