Towards unification of quark and lepton flavors in $A_4$ modular invariance

Abstract

We study quark and lepton mass matrices in the $A_4$ modular symmetry towards the unification of the quark and lepton flavors. We adopt modular forms of weights $2$ and $6$ for quarks and charged leptons, while we use modular forms of weight $4$ for the neutrino mass matrix which is generated by the Weinberg operator. We obtain the successful quark mass matrices, in which the down-type quark mass matrix is constructed by modular forms of weight $2$, but the up-type quark mass matrix is constructed by modular forms of weight $6$. Two regions of $\tau$ are consistent with observed CKM matrix elements. The one is close to $\tau =i$ and the other is in the larger $\mathrm{Im}[\tau ]$. On the other hand, lepton mass matrices work well only at nearby $\tau =i$, which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common $\tau$ region for quarks and leptons, the predicted sum of neutrino masses is $87$--$120$meV taking account of its cosmological bound. Since both the Dirac CP phase ${\delta }_{CP}^{\ell }$ and ${\sin }^2{\theta }_{23}$ are correlated with the sum of neutrino masses, improving its cosmological bound provides crucial tests for our scheme as well as the precise measurement of ${\sin }^2{\theta }_{23}$ and ${\delta }_{CP}^{\ell }$. The effective neutrino mass of the $0\nu \beta \beta$ decay is $⟨m_{ee}⟩=15$--$31$\,meV. It is remarked that the modulus $\tau$ is fixed at nearby $\tau =i$ in the fundamental domain of SL$(2,Z)$, which suggests the residual symmetry $Z_2$ in the quark and lepton mass matrices. The inverted hierarchy of neutrino masses is excluded by the cosmological bound of the sum of neutrino masses.

Comments: 19 pages, 6 tables, 9 figures; the quark and lepton sectors have been significantly improved, and final results have been also changed