$Sp(6,Z)$ modular symmetry in flavor structures: quark flavor models and Siegel modular forms for $\widetilde{\Delta}(96)$

Abstract

We study an approach to construct Siegel modular forms from $Sp(6,Z)$. Zero-mode wave functions on $T^6$ with magnetic flux background behave Siegel modular forms at the origin. Then $T$-symmetries partially break depending on the form of background magnetic flux. We study the background such that three $T$-symmetries $T_I$, $T_{II}$ and $T_{III}$ as well as the $S$-symmetry remain.Consequently, we obtain Siegel modular forms with three moduli parameters $(\omega_1,\omega_2,\omega_3)$, which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for $\widetilde{\Delta}(96)$ in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for $\widetilde{\Delta}(96)$. Around the cusp, $\omega_1=i\infty$, the Siegel modular forms have hierarchical values depending on their $T_I$-charges. We show the deviation of $\omega_1$ from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of $\omega_2$ from imaginary axis.