Rigid invariance as derived from BRS invariance: The abelian Higgs model

TL;DR

This paper derives rigid invariance from BRS invariance in the abelian Higgs model, establishing Ward identities and analyzing the Callan-Symanzik equation to understand symmetry consequences in gauge theories.

Contribution

It constructs rigid and current Ward identities from BRS invariance to all orders in perturbation theory for the abelian Higgs model, aiding understanding of symmetry in gauge theories.

Findings

Rigid invariance leads to scheme-independent relations among Green functions.

The construction of Ward identities is consistent with BRS invariance in spontaneous symmetry breaking.

The Callan-Symanzik equation's form is clarified through this invariance approach.

Abstract

Consequences of a symmetry, e.g.\ relations amongst Green functions, are renormalization scheme independently expressed in terms of a rigid Ward identity. The corresponding local version yields information on the respective current. In the case of spontaneous breakdown one has to define the theory via the BRS invariance and thus to construct rigid and current Ward identity non-trivially in accordance with it. We performed this construction to all orders of perturbation theory in the abelian Higgs model as a prelude to the standard model. A technical tool of interest in itself is the use of a doublet of external scalar ``background'' fields. The Callan-Symanzik equation has an interesting form and follows easily once the rigid invariance is established.