Models with an extended scalar electroweak sector can have vanishing vacuum expectation values in a basis where an underlying symmetry is imposed. Such extensions are very well motivated. If a symmetry prevents couplings between fermions and additional scalars, such scalars could become viable dark matter candidates if some additional criteria are satisfied. We catalogue 𝑆 3 -symmetric three-Higgs-doublet models, also allowing for softly broken 𝑆 3 -symmetric scalar potential terms, based on whether a specific model could possibly accommodate a dark matter candidate. The variety of the 𝑆 3 -symmetric family models arises due to different possibilities to arrange vacuum expectation values. Such models can have vacua with one or two vanishing vacuum expectation values. In our study we assume that the dark matter candidate is stabilised by the Z 2 symmetry. The Z 2 symmetry is a remnant of 𝑆 3 symmetry which survived spontaneous symmetry breaking, and not superimposed over 𝑆 3 . We explore two models; with an without CP violation. These models have a single dark and two active scalar sectors. The active sectors behave in many aspects like a Type-I two-Higgs-doublet model. The dark matter candidate masses, in two cases, are different from the known (previously studied) models with three scalar doublets. After investigating the models in detail, identifying parameters compatible with both theoretical and experimental constraints, we found that the dark matter candidate mass could be within the range of [52.5, 89] GeV or [6.5, 44.5] GeV for a model with 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 Standard Model. 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 Standard Model imposing the existence of a scalar basis where all couplings are real is a sufficient condition for CP to be explicitly conserved. We discuss a three-Higgs-doublet model with an underlying S 3 symmetry, allowing in principle for complex couplings. In this framework it is possible to have either spontaneous or explicit CP violation in the scalar sector, depending on the regions of parameter space corresponding to the different possible vacua of the S 3 symmetric potential. We list all possible vacuum structures allowing for CP violation in the scalar sector specifying whether it can be explicit or spontaneous. It is by now established that CP is violated in the flavour sector and that the Cabibbo-Kobayashi-Maskawa matrix is complex. 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 S 3 group. This classification is based strictly on the exact S 3 -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.

Models with an extended scalar electroweak sector can have vanishing vacuum expectation values. Such behaviour is a result of an underlying symmetry. If a symmetry prevents couplings between fermions and additional scalars, such scalars could become viable dark matter candidates if some additional criteria are satisfied. We catalogue 𝑆 3 -symmetric three-Higgs-doublet models, also allowing for softly broken 𝑆 3 -symmetric scalar potential terms, based on whether a specific model could possibly accommodate a dark matter candidate. The variety of the 𝑆 3 -symmetric family models arises due to different possibilities to arrange vacuum expectation values. Such models can have vacua with one or two vanishing vacuum expectation values. In our study we assume that the dark matter candidate is stabilised by the Z 2 symmetry. The Z 2 symmetry is a remnant of 𝑆 3 symmetry which survived spontaneous symmetry breaking, and not superimposed over 𝑆 3 . We explore two models; with an without CP violation. These models have a single dark and two active scalar sectors. The active sectors behave in many aspects like a Type-I two-Higgs-doublet model. The dark matter candidate masses, in two cases, are different from the known models with three scalar doublets. After investigating the models in detail, identifying parameters compatible with both theoretical and experimental constraints, we found that the dark matter candidate mass could be within the range of [52.5, 89] GeV or [6.5, 44.5] GeV for a model with CP violation.

Models with an extended scalar electroweak sector are well motivated. Such models could accommodate a dark matter candidate if there is an additional scalar representation with a vanishing vacuum expectation value and, in addition, there are no couplings between fermions and the dark matter candidate. The most natural way to have these conditions implemented is to consider models where an underlying symmetry is imposed. Governed by this, we consider a three-Higgs-doublet model with an 𝑆 3 symmetry. Within this framework there are different implementations which could possibly accommodate a dark matter candidate. The family of 𝑆 3 -symmetric three-Higgsdoublet implementations arises due to different vacua and, as a result, different minimisation conditions. In this framework the dark matter candidate falls into the class of weakly interacting massive particles. The dark matter candidate is associated with an Z 2 symmetry which survives spontaneous symmetry breaking and is a remnant of the 𝑆 3 symmetry. We explore two cases, they share many aspects of the Type-I two-Higgs-doublet model plus an inert SU(2) doublet. The main difference between these two cases is the presence of an irremovable phase, which leads to CP violation in one of the implementations. The two candidate cases differ from other previously studied models with three scalar doublets by the fact that they do not allow for heavy dark matter candidates, O (500) GeV. Valid dark matter regions were identified as 𝑚 DM ∈ [52.5, 89] GeV for a model without CP violation and 𝑚 DM ∈ [6.5, 44.5] GeV for a model with CP violation. In the present work we refine the parameter space by applying additional checks to our previous work coming from LHC data and from indirect detection data.8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022