When the C in CP does not matter: anatomy of order-4 CP eigenstates and their Yukawa interactions

We explore the origin and Yukawa interactions of the scalars with peculiar CP-properties which were recently found in a multi-Higgs model based on an order-4 CP symmetry. We relate the existence of such scalars to the enhanced freedom of defining CP, even beyond the well-known generalized CP symmetries, which arises in models with several zero-charge scalar fields. We also show that despite possessing exotic CP quantum numbers, these scalars do not have to be inert: they can have CP-conserving Yukawa interactions provided the CP acts on fermions by also mixing generations. This paper focuses on formal aspects---exposed in a pedagogical manner---and includes a brief discussion of possible phenomenological consequences.