标题： 来自有限模对称性的大尺度和小尺度层次结构 显示英文标题

标题： Large and small hierarchies from finite modular symmetries

摘要： 我们研究了有限模群对称性下由模相关向量状质量引起的辐射修正导致的模稳定化问题。辐射稳定化机制可以将有限模群对称性下的模 $\tau$ （$\Gamma_N$ ($N \in \mathbb{N}$）稳定在 $\mathrm{Im}\,\tau \gg 1$，其中平移对称性 $\tau \to \tau+1$ 大致保持未破缺。平移对称性可被视为实现Froggatt-Nielsen机制的剩余 $\mathbb{Z}_N$ 对称性，其层次参数为 $e^{- 2\pi \mathrm{Im}\,\tau/N} \ll 1$。在这项工作中，我们研究了多个模场的稳定化，从而使模型中各种层次的模形式共存。 例如，一个模数在 $\mathrm{Im}\,\tau_1 \sim 3$ 处稳定下来，负责标准模型中夸克和轻子的分层结构，而另一个在 $\mathrm{Im}\,\tau_2 \sim 15$ 处稳定下来的模数可以解释可能被识别为 QCD 轴子的 $\mathrm{Re}\,\tau_2$ 方向的平坦性。 显示英文摘要

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 $\tau$ 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.