It is by now established that the Standard Model (SM) of Particle Physics cannot be the final theory. In fact, on the one hand the SM leaves several questions open and on the other hand there is already clear evidence for physics beyond the SM. The phenomenon of neutrino oscillations requires the extension of the leptonic sector of the SM. Accounting for the observed baryon asymmetry of the Universe requires new sources of CP violation [1]. Furthermore, there are by now several experimental anomalies in the flavour sector hinting at the existence of new physics [2, 3].
There are strong motivations to consider multi-Higgs extensions of the SM, despite the fact that the properties of the Higgs boson discovered in 2012 are still in experimental agreement with the SM Higgs predictions [4, 5]. Among these motivations are the possibility of having new sources of CP violation. CP violation plays a very important role in nature with implications both for Particle Physics and for Cosmology. Accounting for the observed matter anti-matter asymmetry of the Universe requires the existence of new sources of CP violation beyond the SM. In models with an extended scalar sector CP violation can emerge either explicitly, i.e., at the Lagrangian level, or spontaneously. Spontaneous CP violation occurs in the framework of the electroweak symmetry breaking whenever the Lagrangian conserves CP and the vacuum breaks it. This requires that not all vacuum expectation values be real. In the context of multi-Higgs extensions of the SM imposing the existence of a scalar basis where all couplings are real is a sufficient condition for CP to be explicitly conserved. In the case of spontaneous CP breaking the complex parameters of the Cabibbo-Kobayashi-Maskawa matrix would be generated through phases of the vacuum expectation values of Higgs fields.
Among the simplest extensions of the SM are the so-called two-Higgs-doublet models (2HDM) [6], in which a second SU(2) scalar doublet with the same quantum numbers as the one of the SM is introduced, allowing for explicit CP violation in the scalar sector. The CP properties and cosmological implications of 2HDM have been extensively studied and are still of great interest. Despite the rich phenomenology of 2HDM, models with three Higgs doublets are attracting a lot of interest in the literature. Within three-Higgs-doublet models (3HDM) it is possible to have an additional source of CP violation in the scalar sector while at the same time having natural flavour conservation [7]. The predictability of models with several SU(2) scalar doublets can be quickly lost due to the rapid growth of free parameters as the number of doublets increases. Therefore, it is essential to control the number of free parameters by means of symmetries [8]. It was found that in many cases imposing additional symmetries to multi-Higgs models eliminates the possibility of having CP violation in the Higgs sector.
We study the CP properties of the S3-symmetric 3HDM allowing for complex couplings in the potential. The case of real couplings was studied before in Ref. [9]. With real couplings CP is explicitly conserved by the scalar potential and there is only the possibility of having spontaneous CP violation for special vacua. Different vacua correspond to different regions of parameter space which are determined by imposing the minimisation conditions. In the case of complex couplings some of the CP conserving vacua structures of the real potential now correspond to regions of parameter space that allow for explicit CP violation. For instance, CP can be explicitly violated even in the case where the vacuum preserves the S3 symmetry.
In order to understand what are the possible sources of CP violation in the Yukawa sector we analyse the implications of the different available choices of representations for the quarks under the S3 group. This classification is based strictly on the exact S3-symmetric scalar potential with no soft symmetry breaking terms. The scalar sector of one such model was explored numerically. After applying the theoretical and the most important experimental constraints the available parameter space is shown to be able to give rise to light neutral scalars at the O(MeV) scale.