Large and small hierarchies from finite modular symmetries

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 τ of the finite modular symmetry Γ_N (N ∈ ℕ) at Im τ ≫ 1, where the shift symmetry τ → τ + 1 remains unbroken approximately. The shift symmetry can be considered as the residual ℤ_N symmetry which realizes the Froggatt-Nielsen mechanism with the hierarchy parameter e^{−2πIm τ/N} ≪ 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 Im τ₁ ∼ 3 is responsible for the hierarchical structure of the quarks and leptons in the Standard Model, and another modulus stabilized at Im τ₂ ∼ 15 can account for the flatness of the Re τ₂ direction which may be identified as the QCD axion.