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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 rightarrow Z_2 and S_3 rightarrow Z_3 (or equivalently in the sequel S_3 rightarrow Z_2 rightarrow 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 rightarrow Z_3 breaking.
We construct a 3-3-1 model based on family symmetry S_4 responsible for the neutrino and quark masses. The tribimaximal neutrino mixing and the diagonal quark mixing have been obtained. The new lepton charge mathcal{L} related to the ordinary lepton
We construct a 3-3-1 model based on non-Abelian discrete symmetry $T_7$ responsible for the fermion masses. Neutrinos get masses from only anti-sextets which are in triplets $underline{3}$ and $underline{3}^*$ under $T_7$. The flavor mixing patterns
We present a detailed discussion of the triplet anti-triplet symmetry in 3-3-1 models. The full set of conditions to realize this symmetry is provided, which includes in particular the requirement that the two vacuum expectation values of the two sca
A new S3 flavor model based on $mathrm{SU}(3)_C otimes mathrm{SU}(3)_L otimes mathrm{U}(1)_X$ gauge symmetry responsible for fermion masses and mixings different from our previous work is constructed. The new feature is a two - dimensional represen
The $D_4$ flavor model based on $mathrm{SU}(3)_C otimes mathrm{SU}(3)_L otimes mathrm{U}(1)_X$ gauge symmetry that aims at describing quark mass and mixing is updated. After spontaneous breaking of flavor symmetry, with the constraint on the Higgs va