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. 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 _{CP}^ell and sin ^2theta _{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 ^2theta _{23} and delta _{CP}^ell . The effective neutrino mass of the 0nu beta beta decay is langle m_{ee}rangle =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 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.
Highlights
Many models have been proposed by using the nonAbelian discrete groups S3, A4, S4, A5 and other groups with larger orders to explain the large neutrino mixing angles
If flavors of quarks and leptons are originated from a same two-dimensional compact space, quarks and leptons have the same flavor symmetry and the same value of the modulus τ
We use modular forms of weight 4 for the neutrino mass matrix, which is generated by the Weinberg operator
Summary
Many models have been proposed by using the nonAbelian discrete groups S3, A4, S4, A5 and other groups with larger orders to explain the large neutrino mixing angles. The A4 flavor model is an attractive one because the A4 group is the minimal one including a triplet irreducible representation, which allows for a natural explanation of the existence of three families of leptons [14,15,16,17,18,19,20]. A new approach to the lepton flavor problem appeared based on the invariance under the modular group [21], where the model of the finite modular group 3 A4 has been presented. This work inspired further studies of the modular invariance approach to the lepton flavor problem. It should be emphasized that there is a significant difference between the models based on the A4 modular symmetry and those based on the usual non-Abelian discrete A4 flavor symmetry. The proposed flavor models with modular symmetries. Phenomenological discussions of the neutrino flavor mixing have been done based on A4 [26,27], S4 [28,29], A5 [30], and T [31]
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