Quark hierarchical structures in modular symmetric flavor models at level 6

Abstract

We study modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms, and modular forms in the vicinity of the modular fixed points become hierarchical depending on their residual charges. Thus modular symmetric flavor models in the vicinity of the modular fixed points have a possibility to describe mass hierarchies without fine-tuning. Since describing quark hierarchies without fine-tuning requires ${Z}_{n}$ residual symmetry with $n\ensuremath{\ge}6$, we focus on ${\mathrm{\ensuremath{\Gamma}}}_{6}$ modular symmetry in the vicinity of the cusp $\ensuremath{\tau}=i\ensuremath{\infty}$ where ${Z}_{6}$ residual symmetry remains. We use only modular forms belonging to singlet representations of ${\mathrm{\ensuremath{\Gamma}}}_{6}$ to make our analysis simple. Consequently, viable quark flavor models are obtained without fine-tuning.