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. The viable region of tau is close to tau =i. Lepton mass matrices also work well 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 CPl and sin2 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 sin2 theta 23 and delta CPl. The effective neutrino mass of the 0 nu beta beta decay is mee=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 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.

Automated summary

The study focuses on quark and lepton mass matrices using the A4 modular symmetry to unify quark and lepton flavors. Successful results were obtained for both quark and lepton mass matrices in a specific region, highlighting the importance of the research scheme and the need for verification of predictions.