General Discussions on the SU(2) Vector Boson Dark Matter Model with a Single Higgs Multiplet -- Lagrangian, Discrete Subgroups, and Scalar Classifications

TL;DR

This paper classifies all possible discrete subgroups and scalar configurations in an SU(2) gauge model with a single Higgs multiplet, providing algorithms for operator enumeration and VEV analysis.

Contribution

It systematically enumerates discrete subgroups and scalar representations for SU(2) gauge models with a single Higgs multiplet, offering new tools for model classification.

Findings

Enumerated all discrete subgroups up to dimension 21.

Provided algorithms for operator and VEV enumeration.

Detailed classification of scalar representations within the Higgs multiplet.

Abstract

The vector boson dark matter particles which stem from some broken gauge symmetries usually requires some unbroken symmetries to keep themselves stable. In the previous literature, some simplest cases have been discussed, in which the unbroken symmetry is provided by a remnant subgroup of the gauge group. It would be interesting to ask whether all the possible remnant subgroups as well as all the possible coupling forms can be enumerated. Classifying all the Higgs components into different mass degenerate representations to simplify the diagonalization processes is also necessary. Rather than the ambitious target of providing a general solution to all kinds of gauge groups configured with all forms of the Higgs multiplets, in this paper, we concentrate on the case of $\text{SU(2)}_{\text{D}}$ gauge group together with a single Higgs multiplet. We enumerate all possible discrete subgroups that can survive up to $n=21$, where $n$ is the dimension of the Higgs multiplet. We also provide the general algorithms to enumerate all possible renormalizable operators, to write down the general forms of the vacuum expectation value (VEV) configurations, and to give the detailed results of all the mass degenerate irreducible representations embedded in the Higgs multiplet.