(Bosonic)Mass Meets (Extrinsic)Curvature

TL;DR

This paper provides a geometric perspective on spontaneous symmetry breaking in gauge theories, interpreting bosonic masses and the Higgs mechanism through bundle sections and invariance groups, with applications to the electroweak model.

Contribution

It introduces a geometric framework for understanding mass matrices, symmetry breaking, and the Higgs mechanism, emphasizing the role of vacuum sections and eigenbundle decompositions.

Findings

Mass matrices are constant and independent of vacuum choice.

Decomposition into eigenbundles models particle multiplets.

The unitary gauge has a purely geometric interpretation.

Abstract

In this paper we discuss the mechanism of spontaneous symmetry breaking from the point view of vacuum pairs, considered as ground states of a Yang-Mills-Higgs gauge theory. We treat a vacuum as a section in an appropriate bundle that is naturally associated with a minimum of a (general) Higgs potential. Such a vacuum spontaneously breaks the underlying gauge symmetry if the invariance group of the vacuum is a proper subgroup of the gauge group. We show that each choice of a vacuum admits to geometrically interpret the bosonic mass matrices as ``normal'' sections. The spectrum of these sections turns out to be constant over the manifold and independent of the chosen vacuum. Since the mass matrices commute with the invariance group of the chosen vacuum one may decompose the Hermitian vector bundles which correspond to the bosons in the eigenbundles of the bosonic mass matrices. This decomposition is the geometrical analogue of the physical notion of a ``particle multplet''. In this sense the basic notion of a ``free particle'' also makes sense within the geometrical context of a gauge theory, provided the gauge symmetry is spontaneously broken by some vacuum. We also discuss the Higgs-Kibble mechanism (``Higgs Dinner'') from a geometrical point of view. It turns out that the ``unitary gauge'', usually encountered in the context of discussing the Higgs Dinner, is of purely geometrical origin. In particular, we discuss rotationally symmetric Higgs potentials and give a necessary and sufficient condition for the unitary gauge to exist. As a specific example we discuss in some detail the electroweak sector of the standard model of particle physics in this context.