General discussions on the SU(2) vector boson dark matter model with a single Higgs multiplet: Lagrangian, discrete subgroups, and scalar classifications

The vector boson dark matter particles which stem from some broken gauge symmetries usually require some unbroken symmetries to keep them stable. In 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 the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>SU</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mo stretchy="false">)</a:mo></a:mrow><a:mrow><a:mi mathvariant="normal">D</a:mi></a:mrow></a:msub></a:mrow></a:math> gauge group together with a single Higgs multiplet. We enumerate all possible discrete subgroups that can survive up to <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>n</f:mi><f:mo>=</f:mo><f:mn>21</f:mn></f:math>, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>n</h:mi></h:math> is the dimension of the Higgs multiplet. We also provide the general algorithms to enumerate all possible renormalizable operators, to write 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.