Bulk Quantization of Gauge Theories: Confined and Higgs Phases

TL;DR

This paper presents a 5-dimensional approach to gauge fixing in Yang-Mills theories that maintains positivity of the probability density and unifies confinement and Higgs phases, with implications for numerical simulations.

Contribution

It introduces a 5D formulation replacing traditional gauge fixing, ensuring positivity and BRST control, and connects it with the Faddeev-Popov method and lattice gauge theory.

Findings

The 5D formulation enforces the Gribov region restriction locally.

Ghosts decouple, leading to positive Euclidean probability density.

The Schwinger-Dyson equations match between formulations order by order.

Abstract

We deepen the understanding of the quantization of the Yang-Mills field by showing that the concept of gauge fixing in 4 dimensions is replaced in the 5-dimensional formulation by a procedure that amounts to an $A$-dependent gauge transformation. The 5-dimensional formulation implements the restriction of the physical 4-dimensional gluon field to the Gribov region, while being a local description that is under control of BRST symmetries both of topological and gauge type. The ghosts decouple so the Euclidean probability density is everywhere positive, in contradistinction to the Faddeev-Popov method for which the determinant changes sign outside the Gribov region. We include in our discussion the coupling of the gauge theory to a Higgs field, including the case of spontaneously symmetry breaking. We introduce a minimizing functional on the gauge orbit that could be of interest for numerical gauge fixing in the simulations of spontaneously broken lattice gauge theories. Other new results are displayed, such as the identification of the Schwinger-Dyson equation of the five dimensional formulation in the (singular) Landau gauge with that of the ordinary Faddeev-Popov formulation, order by order in perturbation theory.