Flavor mixing and $CP$ violation from the interplay of an $S_4$ modular group and a generalized $CP$ symmetry

We have performed a systematical analysis of lepton and quark mass models based on ${\mathrm{\Gamma }}_4\cong S_4$ modular symmetry with generalized $CP$ symmetry. We considered both cases; neutrinos are Majorana particles and Dirac particles. All possible nontrivial representation assignments of matter fields are considered, and the most general form of fermion mass matrices are given. The phenomenologically viable models with the lowest number of free parameters together with the results of fit are presented. We find out nine lepton models with seven real free parameters (including the real and imaginary parts of modulus for Majorana neutrinos) which can accommodate the lepton masses and neutrino oscillation data. The prediction for leptogenesis is studied in an example lepton model. The observed baryon asymmetry as well as lepton masses and mixing angles can be explained. For Dirac neutrinos, four lepton models with five real free couplings are compatible with the experimental data. Ten quark models containing seven couplings are found to be able to accommodate the hierarchical quark masses and mixing angles and the $CP$ violation phase. Furthermore, the $S_4$ modular symmetry can provide a unified description of lepton and quark flavor structure, and a benchmark model is presented.

Figure 1

The correlations among the input parameters, lepton mixing angles, $CP$ violation phases and neutrino masses in the model L1. The lepton masses and mixing angles are required to lie in the experimentally preferred $3\sigma$ regions [75]. Notice that the transformation $\tau \to -{\tau }^{*}$ leaves all observables unchanged except shifting the signs of the $CP$ phases ${\delta }_{CP}^l$, ${\alpha }_{21}$ and ${\alpha }_{31}$; consequently we don’t show the $CP$ conjugate region. In the last panel, the red (blue) dashed lines indicate the most general allowed regions for inverted ordering (normal ordering) neutrino mass spectrum respectively, where the neutrino oscillation parameter are varied over their $3\sigma$ ranges. The present upper limit $m_{\beta \beta }<(61–165)\mathrm{meV}$ from KamLAND-Zen [84] is shown by horizontal gray band. The vertical gray exclusion band denotes the current bound coming from the cosmological data of ${\sum }_i{m}_i<0.120\mathrm{eV}$ at 95% confidence level obtained by the Planck Collaboration [80].

Figure 2

The correlations among the input parameters, lepton mixing angles, $CP$ violation phases, and neutrino masses in model L6, where we adopt the same convention as Fig. 1.

Figure 3

The contour plot $Y_B$ in the $\mathrm{Re}⟨\tau ⟩$–$\mathrm{Im}⟨\tau ⟩$ plane. The green region represent the region for which both lepton masses and lepton mixing angles are compatible with experimental data at the $3\sigma$ level or better.

Figure 4

The correlation between $Y_B$ and ${\delta }_{CP}$. The horizonal light blue band denotes the 95% C.L. region of $Y_B$, and the vertical light green band represents the $3\sigma$ range of ${\delta }_{CP}^l$.

Figure 5

The experimentally favored values of $\tau$ are displayed in the first row in the quark-lepton unification model. The quark (lepton) masses and mixing parameters are compatible with experimental data at $3\sigma$ level or better in the light blue (red) area, and the common values of $\tau$ are indicated with black. The second and the third rows are for the correlation among the neutrino masses and mixing parameters.