Natural solution of SUSY $\hat{I}¼$ problem from modulus stabilization in modular flavor model

Abstract

We propose a solution to the SUSY $\hat{I}¼$-problem within the framework of modular flavor symmetry. The explicit $\hat{I}¼$-term is prohibited by modular symmetry, and an effective $\hat{I}¼$-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 ${\text{X}}_{nl}$ superfield. A natural small $\hat{I}{¼}_{eff}$, 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 $\ddot{I}‰$, or from the combined effects of suppression by powers of $q^{1/24}$ [or $(2\Im \ddot{I}„{)}^{-1}$] 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 $\hat{I}{¼}_{eff}$ can also be achieved by a weighton-like mechanism for $H_u{H}_d$ bilinear.

Version accepted in PRD; major changes, discussions added; 40 pages, 2 figures