The 3-3-1 model with inert scalar triplet

We show that the typical 3-3-1 models are only self-consistent if they contain interactions explicitly violating the lepton number. The 3-3-1 model with right-handed neutrinos can by itself work as an economical 3-3-1 model as a natural recognition of the above criteria while it also results an inert scalar triplet (\eta) responsible for dark matter. This is ensured by a Z_2 symmetry (assigned so that only \eta is odd while all other multiplets which perform the economical 3-3-1 model are even), which is not broken by the vacuum. The minimal 3-3-1 model can provide a dark matter by a similar realization. Taking the former into account, we show that the dark matter candidate (H_\eta) contained in \eta transforms as a singlet in effective limit under the standard model symmetry and being naturally heavy. The H_\eta relic density and direct detection cross-section will get right values when the H_\eta mass is in TeV range as expected. The model predicts the H_\eta mass m_{H_\eta}=\lambda_5\times 2 TeV and the H_\eta-nucleon scattering cross-section \sigma_{H_\eta-N}=1.56\times 10^{-44} cm^2, provided that the new neutral Higgs boson is heavy enough than the dark matter.