A realistic pattern of lepton mixing and masses from $$S_4$$ S 4 and CP

We present a supersymmetric model with the flavour symmetry $$S_4 \times Z_3$$ and a CP symmetry which are broken to a $$Z_3$$ subgroup of the flavour symmetry in the charged lepton sector and to $$Z_2 \times $$ CP ( $$\times Z_3$$ ) in the neutrino one at leading order. This model implements an approach capable of predicting lepton mixing angles and Dirac as well as Majorana phases in terms of one free parameter. This parameter, directly related to the size of the reactor mixing angle $$\theta _{13}$$ , can be naturally of the correct order in our model. Atmospheric mixing is maximal, while $$\sin ^2 \theta _{12} \gtrsim 1/3$$ . All three phases are predicted: the Dirac phase is maximal, whereas the two Majorana phases are trivial. The neutrino mass matrix contains only three real parameters at leading order and neutrino masses effectively only depend on two of them. As a consequence, they have to be normally ordered and the absolute neutrino mass scale and the sum of the neutrino masses are predicted. The vacuum of the flavons can be correctly aligned. We study subleading corrections to the leading order results and show that they are small.