Renormalizable theories with symmetry breaking

TL;DR

This paper generalizes the framework of symmetry breaking in renormalizable theories using a geometric approach, demonstrating the validity of Ward identities and current algebra in the presence of anomalies within the BPHZ renormalization scheme.

Contribution

It extends the symmetry breaking formalism to arbitrary compact Lie groups and field multiplets, establishing Ward identities and current algebra at all perturbation orders.

Findings

Ward identities hold to all orders in renormalized perturbation theory.

Local Ward identities are valid up to the Adler-Bardeen anomaly.

The algebraic structure provides a foundation for studying fully quantized gauge theories.

Abstract

The description of symmetry breaking proposed by K. Symanzik within the framework of renormalizable theories is generalized from the geometrical point of view. For an arbitrary compact Lie group, a soft breaking of arbitrary covariance, and an arbitrary field multiplet, the expected integrated Ward identities are shown to hold to all orders of renormalized perturbation theory provided the Lagrangian is suitably chosen. The corresponding local Ward identity which provides the Lagrangian version of current algebra through the coupling to an external, classical, Yang-Mills field, is then proved to hold up to the classical Adler-Bardeen anomaly whose general form is written down. The BPHZ renormalization scheme is used throughout in such a way that the algebraic structure analyzed in the present context may serve as an introduction to the study of fully quantized gauge theories.