Towards unification of quark and lepton flavors in $A_4$ modular invariance

TL;DR

This paper explores a unified approach to quark and lepton flavor structures using $A_4$ modular symmetry, successfully fitting observed data and predicting neutrino masses and CP phases, with implications for future experiments.

Contribution

It introduces a novel $A_4$ modular symmetry framework with specific modular forms for quarks and leptons, unifying their flavor structures and making testable predictions.

Findings

Quark and lepton mass matrices fit observed CKM and neutrino data.

Predicted sum of neutrino masses is 87-120 meV, consistent with cosmological bounds.

Inverted neutrino mass hierarchy is excluded by cosmological constraints.

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 $6$. Two regions of $\tau$ are consistent with observed CKM matrix elements. The one is close to $\tau=i$ and the other is in the larger ${\rm Im }[\tau]$. On the other hand, lepton mass matrices work well only 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 $\sin^2\theta_{23}$ 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 $\sin^2\theta_{23}$ and $\delta_{CP}^\ell$. The effective neutrino mass of the $0\nu\beta\beta$ 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.