The Standard Model (SM) of particle physics has been extensively tested for a few decades and is the most successful description of Nature. Nearly all theoretical SM predictions have been experimentally verified and the last missing piece, the Higgs boson, was discovered in 2012 [1, 2]. This is undoubtedly a fascinating discovery in the field of particle physics and might be the final missing piece. Nevertheless, there is no experimental verification that it is the only Higgs boson, and it will be tested at the LHC and future colliders. It is widely accepted that the SM is an effective low-energy theory, which is capable of explaining physics at low energies, $\mathcal{O}(100)$ GeV, but fails at high energies. While the SM of particle physics keeps on triumphing, there is a vast amount of both theoretical and experimental phenomena that cannot be resolved within the SM framework, some of which are: neutrino oscillations, asymmetry of matter-antimatter, the strong CP problem, etc. From the cosmological point of view the overall SM situation is daunting as it fails to describe not only gravity and dark energy, but also nearly 85% of the matter in the Universe, which is constituted by Dark Matter (DM). The multi-Higgs models could resolve some of the issues and are commonly invoked when models beyond the SM are constructed. Thus, we propose and are motivated that such extension could potentially solve several problems.
The SM uses the minimal Brout-Englert-Higgs mechanism [3, 4, 5], where a single complex SU(2) doublet is considered. The simplest extension of the SM electroweak sector is the Two-Higgs-Doublet Model (2HDM) [6, 7]. In the 2HDM a second SU(2) doublet is added to the SM-like doublet. Such extension predicts a rich scalar spectrum: two additional neutral states $h_{(1,2)}$, and a charged state $h^{\pm}$. The second SU(2) doublet can be further on constrained to result in a viable DM candidate [8]. There are many possibilities to extended the scalar sector by not only the SU(2) doublet but a general n-tuplet, each with its own advantages. A non-minimal scalar sector is well motivated in both Supersymmetry and Grand Unified Theories, where extension of the scalar sector is inevitable.
With limitless possibilities to extend the scalar sector it is crucial not to oversaturate a model with an endless list of free parameters. Let us consider the most general NHDM model. The number of free real parameters is given by $N_{\text{tot}} = N^2 (N^2 + 3)/2$ [9]. With the number of additional SU(2) doublets, the total amount of free parameters grows rapidly, and a specific model loses predictiveness. Symmetries play an important role in controlling the number of free parameters, therefore increasing the predictability of such extensions. For example, the most general 3HDM scalar potential results in 54 free real parameters. However, if we impose a discrete $S_3$ symmetry [10, 11, 12, 13] ad hoc, the number of free parameters decreases to up to ten. In the multi-Higgs extensions some of the problems of the SM, such as the need for new sources of CP violation, which are required to account for the observed baryon asymmetry, can be addressed. An important feature of multi-Higgs extensions of the SM is the possibility of having spontaneous CP violation. Imposing additional symmetries may eliminate the possibility of having spontaneous CP violation.
During the talk we shall cover the 2HDM and the 3HDM models. In particular, we shall present results of Ref. [14], where implications of a mass degeneracy among scalar states and possible symmetries of the 2HDM scalar potential were presented. Moreover, we shall discuss what happens to CP violation in the scalar sector when symmetries are imposed. Also, we introduce a basic principle of how to construct an $S_3$-symmetric 3HDM and how symmetries can lead to massless states [15, 16].