Algebraic Aspects of the Background Field Method

TL;DR

This paper explores the algebraic properties of the background field method in gauge theories, demonstrating how the background effective action maintains key symmetries and equivalences without relying on power-counting arguments.

Contribution

It introduces an algebraic approach to the background field method, preserving BRST invariance and proving the background equivalence theorem through standard techniques.

Findings

Background effective action satisfies Slavnov-Taylor and Ward identities.

The Legendre transform of the background gauge invariant action yields the same physical amplitudes.

W_{bg} cannot generally be derived from a classical action via the Gell-Mann-Low formula.

Abstract

We discuss some algebraic properties of the background field method. We introduce an extra gauge-fixing term for the background gauge field right at the beginning in the action in such a way that BRST invariance is preserved. The background effective action is considered and it is shown to satisfy both the Slavnov-Taylor identities and the Ward identities. This allows to prove the background equivalence theorem by means of the standard techniques. We show that the Legendre transform W_{bg} of the background gauge invariant action gives the same physical amplitudes as the original one we started with. Moreover, we point out that W_{bg} cannot in general be derived from a classical action by the Gell-Mann-Low formula. Finally, we show that the BRST doublet generated from the background field does not modify the anomaly of the original underlying gauge theory. The proof is algebraic and makes no use of arguments based on power-counting.