$CP$ violation conditions in $N$-Higgs-doublet potentials

Conditions for $CP$ violation in the scalar potential sector of general $N$-Higgs-doublet models are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model 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 $N$-Higgs-doublet model 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.