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 $\varepsilon^p$, where $\varepsilon$ 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 $\varepsilon$. To realize the large mass hierarchy as well as sizable CP violation, multi-moduli are required. We show the mass matrix structures with multi-moduli which are consistent with quark flavor observables including CP phase. We also discuss the origins of the large mass hierarchy and CP violation in such mass matrix structures.