Symmetry breaking, subgroup embeddings and the Weyl group

TL;DR

This paper develops a systematic method to identify Higgs vacuum expectation values that break a symmetry group G to various embedded subgroups H, with explicit formulas and applications to grand unified theories and QCD.

Contribution

It introduces a new systematic approach and explicit formulas for finding Higgs vacua breaking G to subgroups, linked to Weyl group formalism, applicable to grand unified theories and QCD models.

Findings

Explicit formula for vacuum manifold points

Identification of vacuum manifold G/H with linear combinations

Application to adjoint Higgs fields and Weyl groups

Abstract

We present a systematic approach to finding Higgs vacuum expectation values, which break a symmetry G to differently embedded isomorphic copies of a subgroup $H \subset G$ . We give an explicit formula for recovering each point in the vacuum manifold of a Higgs field which breaks G -> H. In particular we systematically identify the vacuum manifold G/H with linear combinations of the vacuum expectation values breaking G -> H_1 -> ... -> H_l. We focus on the most applicable case for current work on grand unified theories in extra dimensional models and low-energy effective theories for quantum chromodynamics. Here the subgroup, H, stabilizes the highest weight of the fundamental representation leading to a simple expression for each element of the vacuum manifold; especially for an adjoint Higgs field. These results are illustrated explicitly for adjoint Higgs fields and clearly linked to the mathematical formalism of Weyl groups. We use the final section to explicitly demonstrate how our work contributes to two contemporary high-energy physics research areas.