We have built a renormalizable $U(1)_X$ model with a $\Sigma (18)\times Z_4$ symmetry, whose spontaneous breaking yields the observed standard model (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 $\Sigma (18)$ 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 (NO) and $\langle m_{ee}\rangle =4.88\times 10^{-2} \, \mathrm{eV}$ for inverted ordering (IO), which are well consistent with the recent experimental limits on neutrinoless double beta decay.

中文翻译：

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基于 Σ(18) 离散对称的 U(1)X 模型中的费米子质量和混合

我们建立了一个具有 $\Sigma (18)\times Z_4$ 对称性的可重整化 $U(1)_X$ 模型，其自发断裂产生了观察到的标准模型 (SM) 费米子质量和费米子混合参数。微小质量的轻活性中微子是由非常重的右手马约拉纳中微子介导的 I 型跷跷板机制产生的。据我们所知，该模型是可重整化 $U(1)_X$ 模型中 $\Sigma (18)$ 风味对称性的第一个实现。我们的模型可以成功拟合夸克和轻子扇区的 SM 费米子质量、费米子混合角和 CP 相。获得的夸克和轻子扇区的物理可观测量值与实验数据一致。我们得到一个有效的中微子质量参数$\langle m_{ee}\rangle=1.51\times 10^{-3}\，