Non-Abelian gauge theories with composite fields in the background field method

TL;DR

This paper develops a systematic approach to analyze non-Abelian gauge theories with composite and background fields using the background field method, deriving Ward identities, gauge dependence, and employing finite field-dependent BRST transformations.

Contribution

It introduces generating functionals for such theories, establishes gauge independence on-shell, and applies the framework to models like Gribov--Zwanziger and 2D gravity with torsion.

Findings

Derived Ward identities for composite and background fields.

Proved on-shell gauge independence of the effective action.

Applied the formalism to specific models like Gribov--Zwanziger and 2D gravity.

Abstract

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green's functions for a Yang--Mills theory with composite and background fields are introduced, including the generating functional of vertex Green's functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and utilized on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background.On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov--Zwanziger theory, including the case of a local BRST-invariant horizon, and by the Volovich--Katanaev model of two-dimensional gravity with dynamical torsion.