Theory of Classical Higgs Fields. II. Lagrangians

TL;DR

This paper develops a theoretical framework for classical gauge theories with spontaneous symmetry breaking, focusing on the role of Higgs fields and the factorization of gauge-invariant Lagrangians through covariant differentials.

Contribution

It introduces a geometric approach to gauge theories with symmetry breaking, showing how matter fields and Lagrangians factorize via principal connections on reduced bundles.

Findings

Gauge G-invariant Lagrangians factor through covariant differentials.

Matter fields with symmetry group H are described by sections of a composite bundle.

The theory provides a geometric interpretation of Higgs fields in gauge theories.

Abstract

We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $P\to X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/H\to X$ are treated as classical Higgs fields. In this theory, matter fields with an exact symmetry group $H$ are described by sections of a composite bundle $Y\to P/H\to X$. We show that their gauge $G$-invariant Lagrangian necessarily factorizes through a vertical covariant differential on $Y$ defined by a principal connection on an $H$-principal bundle $P\to P/H$.