Large and small hierarchies from finite modular symmetries

TL;DR

This paper explores how radiative corrections stabilize multiple moduli in finite modular symmetry models, creating hierarchical structures relevant for particle physics and axion physics.

Contribution

It introduces a mechanism for stabilizing multiple moduli fields at different values, enabling coexistence of hierarchical structures within finite modular symmetry frameworks.

Findings

Stabilization of moduli at large imaginary parts using radiative corrections.

Hierarchical values of moduli can explain fermion mass hierarchies.

One modulus can serve as a QCD axion candidate.

Abstract

We study the moduli stabilization by the radiative corrections due to the moduli dependent vector-like masses invariant under the finite modular symmetry. The radiative stabilization mechanism can stabilize the modulus $\tau$ of the finite modular symmetry $\Gamma_N$ ($N \in \mathbb{N}$) at $\mathrm{Im}\,\tau \gg 1$, where the shift symmetry $\tau \to \tau+1$ remains unbroken approximately. The shift symmetry can be considered as the residual $\mathbb{Z}_N$ symmetry which realizes the Froggatt-Nielsen mechanism with the hierarchy parameter $e^{- 2\pi \mathrm{Im}\,\tau/N} \ll 1$. In this work, we study the stabilization of multiple moduli fields, so that various hierarchical values of the modular forms coexist in a model. For example, one modulus stabilized at $\mathrm{Im}\,\tau_1 \sim 3$ is responsible for the hierarchical structure of the quarks and leptons in the Standard Model, and another modulus stabilized at $\mathrm{Im}\,\tau_2 \sim 15$ can account for the flatness of the $\mathrm{Re}\,\tau_2$ direction which may be identified as the QCD axion.