CP violation conditions in N-Higgs-doublet potentials
  Conditions for CP violation in the scalar potential sector of general N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a minimum set of conditions for explicit and spontaneous CP violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on CP-odd pseudoscalar invariants. When the potential is CP invariant, the explicit procedure to reach the real CP-basis and the explicit CP transformation can also be obtained. The procedure to find the real basis and the conditions for CP violation are then extended to general NHDM potentials. The analysis becomes more involved and only a formal procedure to reach the real basis is found. Necessary conditions for CP invariance can still be formulated in terms of group invariants: the CP-odd generalized pseudoscalars. The problem can be completely solved for three Higgs-doublets. 
C. C. Nishi