Phenomenological Consequences of See-Saw in S4 Based Models

It was proposed a flavour model based on the symmetry group S4, managing to describe fermion masses and mixings. The Weinberg operator has been used in order to provide the smallness of the neutrino masses, while a set of scalar fields, getting non-vanishing vacuum expectation values, spontaneously breaks down S4 and provides the Tri-Bimaximal pattern as the lepton mixing matrix. Restricting to this setting, in this paper we analyze possible origins for the effective terms: the type I See-Saw mechanism is the best known approach, but also the type II and III are discussed. The phenomenology related to these models is various and the next future experiments could in principle discriminate among these proposals. Furthermore, we compare our realizations to two relevant A4 based models, also predicting the Tri-Bimaximal lepton mixing, and we find that an analysis on the neutrinoless double beta decay parameters could distinguish among all these realizations. Furthermore a combined measurement of the effective mass and of the lightest neutrino mass could indicate in the next future which is the preferred flavour symmetry group. The introduction of new physics beyond the Standard Model, like heavy right-handed neutrinos, scalar triplets and fermion triplets, let us investigate on leptogenesis and this provides constraints in the realization of the models.