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

Hong Jie Fan, Fei Wang, Ying Kai Zhang

Abstract

We propose a solution to the SUSY $μ$-problem within the framework of modular flavor symmetry. The explicit $μ$-term is prohibited by modular symmetry, and an effective $μ$-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 $μ_{eff}$, 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 $ω$, or from the combined effects of suppression by powers of $q^{1/24}$ [or $(2\Imτ)^{-1}$] 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 $μ_{eff}$ can also be achieved by a weighton-like mechanism for $H_uH_d$ bilinear.

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

Abstract

We propose a solution to the SUSY -problem within the framework of modular flavor symmetry. The explicit -term is prohibited by modular symmetry, and an effective -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 superfield. A natural small , 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 , or from the combined effects of suppression by powers of [or ] along with the asymptotic suppression behavior of typical modular forms away from the fixed point , taking the form of appropriate power of the tiny deviation parameter. A natural small can also be achieved by a weighton-like mechanism for bilinear.

Paper Structure

This paper contains 13 sections, 107 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Modular $\Gamma_N$ symmetry
  3. Natural $\mu$-term in modular flavor model
  4. Stabilization of modulus field with nilpotent Goldstino ${\textbf{X}}_{nl}$ superfield
  5. The case with $k_X=0$
  6. The case with $k_X=12$
  7. Regeneration of naturally small $\mu_{eff}$ following modulus stabilization
  8. Natural $\mu_{eff}$ with modulus stabilization near the fixed point $\omega\equiv e^{i2\pi/3}$
  9. Natural $\mu_{eff}$ with modulus stabilization at medium $\Im\tau$
  10. Conclusions
  11. Properties of Modular $A_4$
  12. Eisenstein Series
  13. Solutions of $H^\prime(\tau)=0$

Figures (1)

  • Figure 2: The values of $V(X_{nl},\bar{X}_{nl},\tau,{\tau}^*)$ within the fundamental domain in the scenario with $k_X=12$ and $X_{nl}=0$ for various parameter choices when $f_0\neq 0$ and $f_0\gg c_1^2$. We adopt $\mathcal{P}(j)=1$ for the left panel and $\mathcal{P}(j(\tau))=(j(\tau)-j(\tau_0))^2$ with $\tau_0=1.05\times e^{i\frac{\pi}{2}(1+\frac{1}{6})}$ for the right panel.