The SU(3)C ⊗ SU(3)L ⊗ U(1)X gauge model with the minimal scalar sector (two Higgs triplets) is studied in detail. One of the vacuum expectation values u is a source of lepton-number violations and a reason for the mixing among the charged gauge bosons -the standard model W and the bilepton (with L = 2) gauge bosons as well as among neutral non-Hermitian X 0 and neutral gauge bosons: the photon, the Z and the new Z ′ . Because of these mixings, the lepton-number violating interactions exist in both charged and neutral gauge boson sectors. An exact diagonalization of the neutral gauge boson sector is derived and bilepton mass splitting is also given. The lepton-number violation happens only in the neutrino but not in the charged lepton sector. In this model, leptonnumber changing (∆L = ±2) processes exist but only in the neutrino sector. Constraints on VEVs of the model are estimated and u ≃ O(1) GeV, v ≃ v weak = 246 GeV and ω ≃ O(1) TeV.

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Interactions among the standard model gauge bosons and scalar fields in the framework of SU(3)C ⊗ SU(3)L ⊗ U(1)X gauge model with minimal (economical) Higgs content are presented. From these couplings, all scalar fields including the neutral scalar h and the Goldstone bosons can be identified and their couplings with the usual gauge bosons such as the photon, the charged W ± and the neutral Z, without any additional condition, are recovered. In the effective approximation, full content of scalar sector can be recognized. The CP-odd part of Goldstone associated with the neutral non-Hermitian bilepton gauge boson G X 0 is decouple, while its CP-even counterpart has the mixing by the same way in the gauge boson sector.

We propose a new and realistic 3-3-1 model with the minimal lepton and scalar TeV and yield that the third quark generation is different from the first two. The correct mass generation for top quark implies that the minimal scalar sector as proposed is unique. By the simple 3-3-1 model, the other scalars beside the minimal ones can behave as inert fields responsible for dark matter. A triplet, doublet and singlet dark matter are respectively recognized. Our proposals provide the solutions for the long-standing dark matter issue in the minimal 3-3-1 model.

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 L related to the ordinary lepton charge L and a SU (3) charge by L = 2 √ 3 T 8 + L and the lepton parity P l = (−) L known as a residual symmetry of L have been introduced which provide insights in this kind of model. The expected vacuum alignments resulting in potential minimization can origin from appropriate violation terms of S 4 and L. The smallness of seesaw contributions can be explained from the existence of such terms too. If P l is not broken by the vacuum values of the scalar fields, there is no mixing between the exotic and the ordinary quarks at the tree level.