Fermion mass hierarchy in modular flavor symmetry

The modular flavor symmetry is studied to explain the mass hierarchy of quarks and leptons. In this review, at first, we introduce briefly the framework of the modular symmetries and its residual symmetries, and then, present the general framework of generating the mass hierarchy close to the symmetric points of modulus without fine-tuning. Finally, we present a minimal [Formula: see text] modular invariant model of quarks. We then describe how to obtain the hierarchical quark mass patterns in the vicinity of [Formula: see text] and investigate the flavor structure of the quark mass matrices. The results of our study show that describing correctly the quark mass hierarchies without severe fine-tuning and the CP violation is remarkably challenging within the modular invariance approach to flavors.