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

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 ${ϵ}^p$, 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.