Non-holomorphic $S^{\prime}_{4}$ modular symmetry for leptons and leptogenesis
We perform a comprehensive and systematic investigation of lepton models based on the non-holomorphic $S^{\prime}_{4}$ modular symmetry, by using level 4 polyharmonic Maaß forms spanning integer weights from $-4$ to $6$. The light neutrino masses are generated by the type-I seesaw mechanism with two right-handed neutrinos, no flavon fields other than the modulus $τ$ is introduced, and the generalized CP symmetry is not imposed. An exhaustive numerical analysis yields 36 viable models with only four real couplings besides the modulus $τ$ when neutrino masses are normal ordering. They are classified into three categories, each containing twelve models which yield quite similar predictions for lepton observables and are distinguished by the assignment of $E^c_1$. Furthermore, we perform a detailed numerical analysis for one representative model from each category. These representative models are found to yield very sharp predictions for neutrino masses and mixing parameters, and they are distinguished by the predictions for the atmospheric mixing angle $θ_{23}$, the Dirac CP phase $δ_{CP}$ and the Majorana CP phase $α_{21}$. Furthermore, we find that only two of these three representative models accommodate successful thermal leptogenesis in the unflavored regime, reproducing the observed baryon asymmetry with the identical parameter values that satisfy neutrino oscillation data. In these models, the real part of the modulus $τ$ is the unique source of CP violation in both lepton mixing and leptogenesis.