Three-Higgs-doublet model with A4 symmetry

We worked out in detail the three-Higgs-doublet extension of the standard model when the A4 symmetry, which is imposed to solve the flavor problem, is extended to the scalar sector. The three doublets may be related to the fermion mass generation and, in particular, they may be the unique responsible for generating the neutrino masses. If this is the case, the respective VEVs have to be quite smaller than the electroweak scale. The usual hierarchy in the Yukawa couplings is now moved to a hierarchy in the VEVs which may be justified on dynamical grounds. We consider here the mass spectra in the scalar sector in several situations. When the three doublets are those related with neutrino masses some light scalar arise. However, the later fields are safe, from the phenomenological point of view, since they couple mainly to neutrinos.