Abstract
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.
Highlights
The Standard Model (SM) is agnostic about the origin of the CP violation which we observe in weak interactions [1]
From the arguments similar to those described in the Appendix, we conclude that, once again, there is a mismatch between the imposed form of the CP symmetry and the requirement that, when squared, it should generate a rephasing symmetry of order 4
We have constructed here two versions of 5HDM with CP8 and CP16, and the methods we have used can be generalized to CP symmetries of any order 2k 1⁄4 2p, should the need arise
Summary
The Standard Model (SM) is agnostic about the origin of the CP violation which we observe in weak interactions [1]. This feature has clearly visible consequences in the scalar potential: despite the model being CP conserving, it is impossible to find a basis in which all coefficients would be real This is due to the existence of gauge-invariant nonHermitian combinations of Higgs fields which are invariant under CP4 instead of being mapped to their Hermitian conjugates. The five doublets are grouped in a natural way: one-Higgs doublet acquires the vacuum expectation value (vev) and produces the SM-like Higgs particle, while the inert sector includes two pairs of two doublets, with the CP transformation mixing the doublets within each pair When constructing these examples, we will explain the strategy of building models with even higher-order CP symmetries, should an interest in such models appear. We want to convey to the community the message that all these previously overlooked possibilities exist
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