$SU(3)_C\times SU(3)_L\times U(1)_X$ model from $SU(6)$

Abstract: We propose the $SU(3)_C\times SU(3)_L\times U(1)_X$ model arising from $SU(6)$ breaking. One family of the Standard Model (SM) fermions arises from two $\bar{6}$ representations and one $15$ representation of $SU(6)$ gauge symmetry. To break the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry down to the SM, we introduce three $SU(3)_L$ triplet Higgs fields, where two of them come from the $\bar{6}$ representation while the other one from the $15$ representation. We study the gauge boson masses and Higgs boson mass in detail, and find that the vacuum expectation value (VEV) of the Higgs field for $SU(3)_L\times U(1)_X$ gauge symmetry breaking is around 10 TeV. The neutrino masses and mixing can be generated via the littlest inverse seesaw mechanism. In particular, we have normal hierarchy for neutrino masses and the lightest active neutrino is massless. Also, we consider constraints from the charged lepton flavor changing decays as well. Furthermore, introducing two $SU(3)_L$ adjoint fermions, one $SU(3)_C$ adjoint scalar, and one $SU(3)_L$ triplet scalar, we can achieve gauge coupling unification within 1\%. These extra particles can provide a dark matter candidate as well.