Quark mass hierarchies and CP violation in A 4 × A 4 × A 4 modular symmetric flavor models

Abstract

Abstract We study A 4 × A 4 × 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 τ = i∞ and ω, A 4 is broken to Z 3 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 A 4 × A 4 × A 4, the residual symmetry is Z 3 × Z 3 × Z 3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |V CKM | 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 J CP, quark mass ratios and CKM matrix |V CKM | simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal.

Keywords

• CKM Parameters
• CP Violation
• Discrete Symmetries
• Flavour Symmetries