Custodial SO(4) symmetry and CP violation in N-Higgs-doublet potentials

TL;DR

This paper explores how to implement custodial SO(4) symmetry in N-Higgs-doublet models, showing it leads to CP invariance and providing various equivalent formulations, including new ones for 2HDM.

Contribution

It demonstrates that all custodial SO(4) implementations are basis-equivalent under certain conditions and introduces new implementations for the 2HDM.

Findings

Custodial SO(4) implementation is basis-equivalent to a canonical form.

Invariance under SO(4) implies CP invariance of the potential.

New implementation of custodial symmetry for 2HDM not previously considered.

Abstract

We study the implementation of global $SO(4)\sim SU(2)_L\otimes SU(2)_R$ symmetry in general potentials with N-Higgs-doublets in order to obtain models with custodial $SO(3)_C$ symmetry. We conclude that any implementation of the custodial SO(4) symmetry is equivalent, by a basis transformation, to a canonical one if $SU(2)_L$ is the gauge factor, $U(1)_Y$ is embedded in $SU(2)_R$ and we require $N$ copies of the doublet representation of $SU(2)_R$. The invariance by SO(4) automatically leads to a CP invariant potential and the basis of the canonical implementation of SO(4) is aligned to a basis where CP-symmetry acts in the standard fashion. We show different but equivalent implementations for the 2HDM, including an implementation not previously considered.