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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 modelswith custodial $SO(3)_C$ symmetry. We conclude that any implementation of thecustodial SO(4) symmetry is equivalent, by a basis transformation, to acanonical 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 thebasis of the canonical implementation of SO(4) is aligned to a basis whereCP-symmetry acts in the standard fashion. We show different but equivalentimplementations for the 2HDM, including an implementation not previouslyconsidered.

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