Towards unification of quark and lepton flavors in $$A_4$$ modular invariance
AbstractWe study quark and lepton mass matrices in the $$A_4$$ 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 $$\tau $$ τ is close to $$\tau =i$$ τ = i . Lepton mass matrices also work well at nearby $$\tau =i$$ τ = i , which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common $$\tau $$ 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 $$\delta _{CP}^\ell $$ δ CP ℓ and $$\sin ^2\theta _{23}$$ 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 $$\sin ^2\theta _{23}$$ sin 2 θ 23 and $$\delta _{CP}^\ell $$ δ CP ℓ . The effective neutrino mass of the $$0\nu \beta \beta $$ 0 ν β β decay is $$\langle m_{ee}\rangle =15$$ ⟨ m ee ⟩ = 15 –31 meV. It is remarked that the modulus $$\tau $$ τ is fixed at nearby $$\tau =i$$ τ = i in the fundamental domain of SL(2, Z), which suggests the residual symmetry $$Z_2$$ 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.
