Fermion masses and mixings in a $U(1)_X$ model based on the $Σ(18)$ discrete symmetry
We have built a renormalizable $U(1)_X$ model with a $Σ(18)\times Z_4$ symmetry, whose spontaneous breaking yields the observed SM fermion masses and fermionic mixing parameters. The tiny masses of the light active neutrinos are produced by the type I seesaw mechanism mediated by very heavy right handed Majorana neutrinos. To the best of our knowledge, this model is the first implementation of the $Σ(18)$ flavor symmetry in a renormalizable $U(1)_X$ model. Our model allows a successful fit for the SM fermion masses, fermionic mixing angles and CP phases for both quark and lepton sectors. The obtained values for the physical observables of both quark and lepton sectors are in accordance with the experimental data. We obtain an effective neutrino mass parameter of $\langle m_{ee}\rangle=1.51\times 10^{-3}\, \mathrm{eV}$ for normal ordering and $\langle m_{ee}\rangle =4.88\times 10^{-2} \, \mathrm{eV}$ for inverted ordering which are well consistent with the recent experimental limits on neutrinoless double beta decay.