Quark mass hierarchies and CP violation in $A_4\times A_4\times A_4$ modular symmetric flavor models

Abstract

We study $A_4 \times A_4 \times A_4$ modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points $\tau = i\infty$ and $\omega$, $A_4$ is broken to $Z_3$ residual symmetry. When the modulus $\tau$ is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with $A_4\times A_4 \times A_4$, the residual symmetry is $Z_3 \times Z_3 \times Z_3$ which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix $|V_{\textrm{CKM}}|$ without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of $\tau$. We also show possibilities to realize observed values of the Jarlskog invariant $J_{\textrm{CP}}$, quark mass ratios and CKM matrix $|V_{\textrm{CKM}}|$ simultaneously, if $\mathcal{O}(10)$ adjustments in coefficients of Yukawa couplings are allowed.