Multi-Higgs models with $CP$ symmetries of increasingly high order

When building $CP$-symmetric models beyond the Standard Model, one can impose $CP$ symmetry of higher order. This means that one needs to apply the $CP$ transformation more than two times to get the identity transformation, but still the model is perfectly $CP$-conserving. A multi-Higgs-doublet model based on $CP$ symmetry of order 4, dubbed CP4, was recently proposed and its phenomenology is being explored. Here, we show that the construction does not stop at CP4. We build examples of renormalizable multi-Higgs-doublet potentials which are symmetric under CP8 or CP16, without leading to any accidental symmetry. If the vacuum conserves $CP$ symmetry of order $2k$, then the neutral scalars become $CP$ eigenstates, which are characterized not by $CP$ parities but by $CP$ charges defined modulo $2k$. One or more lightest states can be the dark matter candidates, which are protected against decay not by the internal symmetry but by the exotic $CP$. We briefly discuss their mass spectra and interaction patterns for CP8 and CP16.