CP phase in modular flavor models and discrete Froggatt-Nielsen models

Abstract

We study the large mass hierarchy and CP violation in the modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms. Modular forms near the modular fixed points are approximately given by ϵp, where ϵ and p denote the small deviation from the fixed points and their residual charges. Thus, mass matrices have the hierarchical structures depending on the residual charges and have a possibility describing the large mass hierarchy without fine-tuning. Similar structures of mass matrices are also obtained in Froggatt-Nielsen models. Nevertheless, it seems to be difficult to induce a sufficient amount of CP violation by a single small complex parameter ϵ. To realize the large mass hierarchy as well as sizable CP violation, multimoduli are required. We show the mass matrix structures with multimoduli that are consistent with quark flavor observables including the CP phase. We also discuss the origins of the large mass hierarchy and CP violation in such mass matrix structures. Published by the American Physical Society 2024