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 $τ$ are consistent with observed CKM matrix elements. The one is close to $τ=i$ and the other is in the larger ${\rm Im }[τ]$. On the other hand, lepton mass matrices work well only at nearby $τ=i$, which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common $τ$ 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 $δ_{CP}^\ell$ and $\sin^2θ_{23}$ 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 $\sin^2θ_{23}$ and $δ_{CP}^\ell$. The effective neutrino mass of the $0νββ$ decay is $\langle m_{ee}\rangle=15$--$31$\,meV. It is remarked that the modulus $τ$ is fixed at nearby $τ=i$ 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.