Towards unification of quark and lepton flavors in A₄ modular invariance

We study quark and lepton mass matrices in the A₄ 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 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--120meV taking account of its cosmological bound. Since both the Dirac CP phase δ₂₏^ and ²θ₂₃ 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 ²θ₂₃ and δ₂₏^. The effective neutrino mass of the 0νββ decay is m₄₄=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 Z₂ 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.