A class of $S_3$-inspired Three-Higgs-Doublet Models with complex vacuum

In this paper we analyze in detail an S3-symmetric three-Higgs-doublet model with a specific vacuum configuration. This analysis allows us to illustrate important features of models with several Higgs doublets, such as the possibility of having spontaneous CP violation. We start with a real potential and pick a particularly interesting complex vacuum configuration, which does not violate CP before adding soft breaking terms to the potential. We study the role played by different soft symmetry breaking terms. These are essential for our choice of vacuum in order to remove unwanted massless scalars that arise from the spontaneous breaking of an accidental continuous symmetry. We list scalar sector and scalar-gauge sector couplings for the particular case we consider in detail in this work. Results presented in this paper will be useful for model building, in particular for implementations of models with S3 symmetry and spontaneous CP violation, extensions of the fermionic sector with realistic Yukawa couplings and for dark matter studies.