Natural solution of SUSY $\mu$ problem from modulus stabilization in modular flavor model

We propose a solution to the SUSY $\mu$-problem within the framework of modular flavor symmetry. The explicit $\mu$-term is prohibited by modular symmetry, and an effective $\mu$-term is regenerated following the stabilization of the modulus field. We examine the stabilization mechanism of a single modulus field with the presence of SUSY breaking contributions described by the non-linear SUSY realization scheme involving a nilpotent Goldstino $\textbf{X}_{nl}$ superfield. A natural small $\mu_{eff}$, significantly smaller than the SUSY scale, can result from either the expansion of typical modular forms using a small deviation parameter near the fixed point $\omega$, or from the combined effects of suppression by powers of $q^{1/24}$ [or $(2\Im\tau)^{-1}$] along with the asymptotic suppression behavior of typical modular forms away from the fixed point $i\infty$, taking the form of appropriate power of the tiny deviation parameter. A natural small $\mu_{eff}$ can also be achieved by a weighton-like mechanism for $H_uH_d$ bilinear.