Natural Flavour Conservation in a three Higg-doublet Model

We consider an extension of the standard model (SM) with three $SU(2)$ scalar doublets and a discrete $S_3\otimes \mathbb{Z}_2$ symmetries. The irreducible representation of $S_3$ has a singlet and a doublet, and here we show that the singlet corresponds to the SM-like Higgs and the two additional $SU(2)$ doublets forming a $S_3$ doublet are inert. In general, in a three scalar doublet model, with or without $S_3$ symmetry, the diagonalization of the mass matrices implies arbitrary unitary matrices. However, we show that in our model these matrices are of the tri-bimaximal type. We also analyzed the scalar mass spectra and the conditions for the scalar potential is bounded from below at the tree level. We also discuss some phenomenological consequences of the model.