Neutrino mixing with non-zero ${\theta }_{13}$ and CP violation in the 3-3-1 model based on $A_4$ flavor symmetry

We propose a 3-3-1 model with neutral fermions based on $A_4$ flavor symmetry responsible for fermion masses and mixings with non-zero ${\theta }_{13}$. To get realistic neutrino mixing, we just add a new $SU(3{)}_L$ triplet being in $\underset{‾}{3}$ under $A_4$. The neutrinos get small masses from two $SU(3{)}_L$ antisextets and one $SU(3{)}_L$ triplet. The model can fit the present data on neutrino masses and mixing as well as the effective mass governing neutrinoless double beta decay. Our results show that the neutrino masses are naturally small and a little deviation from the tri-bimaximal neutrino mixing form can be realized. The Dirac CP violation phase $\delta$ is predicted to either ${5.41}^{\circ }$ or ${354.59}^{\circ }$ with ${\theta }_{23}\mathrm{\ne }\frac{\pi }{4}$.