Quark and lepton flavors with common modulus $τ$ in $A_4$ modular symmetry

We study quark and lepton mass matrices with the common modulus $τ$ in the $A_4$ modular symmetry. The viable quark mass matrices are composed of modular forms of weights $2$, $4$ and $6$. It is remarked that the modulus $τ$ is close to $i$, which is a fixed point in the fundamental region of SL$(2,Z)$, and the CP symmetry is not violated. Indeed, the observed CP violation is reproduced at $τ$ which is deviated a little bit from $τ=i$. The charged lepton mass matrix is also given by using modular forms of weights $2$, $4$ and $6$, where five cases have been examined. The neutrino mass matrix is generated in terms of the modular forms of weight $4$ through the Weinberg operator. Lepton mass matrices are also consistent with the observed mixing angles at $τ$ close to $i$ for NH of neutrino masses. Allowed regions of $τ$ of quarks and leptons overlap each other for all cases of the charged lepton mass matrix. However, the sum of neutrino masses is crucial to test the common $τ$ for quarks and leptons. The minimal sum of neutrino masses $\sum m_i$ is $140$meV at the common $τ$. The inverted hierarchy of neutrino masses is unfavorable in our framework. It is emphasized that our result suggests the residual symmetry $\mathbb{Z}_2^{S}=\{ I, S \}$ in the quark and lepton mass matrices.

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