Representation-theoretical characterization of canonical custodial symmetry in NHDM potentials

TL;DR

This paper provides a representation-theoretical framework to identify canonical custodial symmetry in N-Higgs-doublet models, offering practical algorithms for potentials with more than two doublets.

Contribution

It introduces necessary and sufficient conditions for custodial symmetry in NHDM potentials using basis-covariant vectors and Lie algebra representations.

Findings

Conditions are computationally simple for N=3,4,5.

Practical algorithms for symmetry detection are developed.

Framework relates custodial symmetry to Lie algebra basis vectors.

Abstract

By considering the basis-covariant constituents of N-Higgs-doublet potentials, we derive necessary and sufficient conditions for canonical SO(4)_C Custodial Symmetry (CS) of potentials with N>2 doublets, based on representation-theoretical and geometrical relations. In essence, our characterization relates the presence of canonical CS to basis-covariant vectors corresponding to particular bases of the defining representation of the orthogonal Lie algebras. For N=3,4 and 5, the conditions demand little computational effort to be evaluated, and we provide practical algorithms that may be efficiently implemented in a computer program, for deciding whether or not a potential is custodial-symmetric.