Classical Higgs fields

TL;DR

This paper explores classical gauge theories with spontaneous symmetry breaking, showing how matter and Higgs fields are modeled within principal bundles and how gauge invariance is maintained despite symmetry reduction.

Contribution

It demonstrates that matter fields with exact symmetry groups can be incorporated into gauge theories with broken symmetries through associated composite bundles and factorization of the Lagrangian.

Findings

Matter fields are represented by sections of composite bundles associated to principal G-bundles.

Gauge invariant Lagrangian factorizes through a covariant differential determined by a principal H-connection.

The approach applies to Cartan decompositions, expressing connections in terms of gauge potentials for broken symmetries.

Abstract

We consider classical gauge theory on a principal bundle P->X in a case of spontaneous symmetry breaking characterized by the reduction of a structure group G of P->X to its closed subgroup H. This reduction is ensured by the existence of global sections of the quotient bundle P/H->X treated as classical Higgs fields. Matter fields with an exact symmetry group H in such gauge theory are considered in the pairs with Higgs fields, and they are represented by sections of a composite bundle Y->P/H->X, where Y->P/H is a fiber bundle associated to a principal bundle P->P/H with a structure group H. A key point is that a composite bundle Y->X is proved to be associated to a principal G-bundle P->X. Therefore, though matter fields possess an exact symmetry group H, their gauge G-invariant theory in the presence of Higgs fields can be developed. Its gauge invariant Lagrangian factorizes through the vertical covariant differential determined by a connection on a principal H-bundle P->P/H. In a case of the Cartan decomposition of a Lie algebra of G, this connection can be expressed in terms of a connection on a principal bundle P->X, i.e., gauge potentials for a group of broken symmetries G.