Canonical Transformations and Renormalization Group Invariance in the presence of Non-trivial Backgrounds

TL;DR

This paper demonstrates that in SU(N) Yang-Mills theory, the classical background-quantum splitting is deformed by a quantum canonical transformation, which determines the background dependence of Green's functions and aids in deriving the renormalization group equation.

Contribution

It introduces a canonical transformation framework that accounts for quantum deformations of background-quantum splitting in gauge theories, applicable at both perturbative and non-perturbative levels.

Findings

Derived the renormalization group equation with background fields.

Calculated the one-loop deformation in an SU(2) instanton background.

Established the background dependence of Green's functions via a canonical transformation.

Abstract

We show that for a SU(N) Yang-Mills theory the classical background-quantum splitting is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity of the theory. This canonical transformation acts on all the fields (including the ghosts) and antifields; it uniquely fixes the dependence on the background field of all the one-particle irreducible Green's functions of the theory at hand. The approach is valid both at the perturbative and non-perturbative level, being based solely on symmetry requirements. As a practical application, we derive the renormalization group equation in the presence of a generic background and apply it in the case of a SU(2) instanton. Finally, we explicitly calculate the one-loop deformation of the background-quantum splitting in lowest order in the instanton background.