Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models

Abstract

We study A4× A4× A4 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 τ = i∞ and ω, A4 is broken to Z3 residual symmetry. When the modulus τ 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 A4× A4× A4, the residual symmetry is Z3× Z3× Z3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |VCKM| without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of τ. We also show possibilities to realize observed values of the Jarlskog invariant JCP, quark mass ratios and CKM matrix |VCKM| simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal.