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

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Large and small hierarchies from finite modular symmetries

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月14日

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  • Tetsutaro Higaki,
  • Junichiro Kawamura,
  • Tatsuo Kobayashi,
  • Kaito Nasu,
  • Riku Sakuma

摘要

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 ΓN (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 ℤN 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.