Quark mass and mixing in the 3-3-1 model with neutral leptons based on $D_4$ flavor symmetry

The $D_4$ flavor model based on $$\mathrm{S}\mathrm{U}$(3)_C \otimes $\mathrm{S}\mathrm{U}$(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 vacuum expectation values (VEVs) in the Yukawa couplings, all of quarks have consistent masses, and a realistic quark mixing matrix can be realized at the first order of perturbation theory.