We propose two 3-3-1 models (with either neutral fermions or right-handed neutrinos) based on $S_3$ flavor symmetry responsible for fermion masses and mixings. The models can be distinguished upon the new charge embedding ($\mathcal{L}$) relevant to lepton number. The neutrino small masses can be given via a cooperation of type I and type II seesaw mechanisms. The latest data on neutrino oscillation can be fitted provided that the flavor symmetry is broken via two different directions $S_3\to Z_2$ and $S_3\to Z_3$ (or equivalently in the sequel $S_3\to Z_2\to \{\mathrm{\text{Identity}}\}$), in which the second direction is due to a scalar triplet and another antisextet as small perturbation. In addition, breaking of either lepton parity in the model with neutral fermions or lepton number in the model with right-handed neutrinos must be happened due to the $\mathcal{L}$-violating scalar potential. The TeV seesaw scale can be naturally recognized in the former model. The degenerate masses of fermion pairs $(\mu ,\tau )$, $(c,t)$ and $(s,b)$ are, respectively, separated due to the $S_3\to Z_3$ breaking.

• Received 28 November 2011

DOI:https://doi.org/10.1103/PhysRevD.85.053001