Towards unification of quark and lepton flavors in A${}_4$ modular invariance

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. The viable region of τ is close to τ=i. Lepton mass matrices also work well 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${}^2$θ${}_{23}$ 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${}^2$θ${}_{23}$ and δ${}_{CP}^{\ell }$. The effective neutrino mass of the 0νββ decay is ⟨m${}_{ee}$⟩ = 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.