Spontaneous CP violation and $\mu -\tau$ symmetry in two-Higgs-doublet models with flavour conservation

In multi-Higgs-doublet models, requiring simultaneously that (i) CP violation only arises spontaneously, (ii) tree level scalar flavour changing couplings are absent and (iii) the fermion mixing matrix is CP violating, can only be achieved in a very specific manner. A general approach with new clarifying insights on the question is presented. Considering the quark sector, that peculiar possibility is not viable on phenomenological grounds. We show that, considering the lepton sector, it is highly interesting and leads to viable models with $\mu -\tau$ symmetric PMNS matrices. Phenomenological implications of the models, both for Dirac and Majorana (in a type I seesaw scenario) neutrinos, are analysed.