Slavnov-Taylor constraints for non-trivial backgrounds

TL;DR

This paper introduces an algebraic method to evaluate Green's functions in SU(N) Yang-Mills theory with non-trivial backgrounds, leveraging Slavnov-Taylor identities to determine background dependence beyond perturbation theory.

Contribution

It presents a novel algebraic procedure for analyzing Green's functions in non-trivial backgrounds using Slavnov-Taylor identities, applicable beyond perturbation theory and suitable for lattice implementations.

Findings

Dependence of vertex functional on background uniquely determined by Slavnov-Taylor identities.

Quantum deformations to background-quantum splitting encoded in a specific 1-PI correlator.

Application demonstrated through analysis of ghost two-point and Kugo-Ojima functions in an instanton background.

Abstract

We devise an algebraic procedure for the evaluation of Green's functions in SU(N) Yang-Mills theory in the presence of a non-trivial background field. In the ghost-free sector the dependence of the vertex functional on the background is shown to be uniquely determined by the Slavnov-Taylor identities in terms of a certain 1-PI correlator of the covariant derivatives of the ghost and the anti-ghost fields. At non-vanishing background this amplitude is shown to encode the quantum deformations to the tree-level background-quantum splitting. The approach only relies on the functional identities of the model (Slavnov-Taylor identities, $b$-equation, anti-ghost equation) and thus it is valid beyond perturbation theory, and in particular in a lattice implementation of the background field method. As an example of the formalism we analyze the ghost two-point function and the Kugo-Ojima function in an instanton background in SU(2) Yang-Mills theory, quantized in the background Landau gauge.