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

Full text: We study the implementation of global SO(4) ∼ SU(2)L ⊗ 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. We can separate our goals in this study into two categories: construction and identification. The firs refers to the construction of general custodial invariant NHDM potentials and analysis of the models. The second goal refers to the methods of identifying the SO(4) symmetry if it is not manifest and concealed such as the CP symmetry acting in a general basis. It is then important to study the interplay between the imposition of SO(4) symmetry and the reparametrization group SU(N)H. In particular, it is important to emphasize that the global SO(4) symmetry, like CP symmetry, is not contained in U(1)Y ⊗ SU(2)L SU(N)H as most of the symmetries usually considered for NHDMs are. (author)