标题： 带$Γ_2$模块不变性的轻子偶极子算符 显示英文标题

标题： Leptonic dipole operator with $Γ_2$ modular invariance

摘要： 我们研究了与μ子异常磁矩最近数据相关的轻子电弱偶极矩和轻子味违反衰变，即$(g-2)_{\mu}$，在SMEFT框架下结合弦假设。 我们采用了Meloni-Parriciatu成功的$\Gamma_2$模块不变模型作为轻子的味对称性。 假设$(g-2)_{\mu}$和$\Delta a_{\mu}$的异常是新物理（NP）的证据，我们将它与电子的异常磁矩$\Delta a_e$、电子电弱偶极矩$d_e$和$\mu\to e \gamma$衰变相关联。 我们发现 NP 对$\Delta a_{e(\mu)}$的贡献与轻子质量平方成正比，同样如朴素标度$\Delta a_\ell \propto m^2_\ell$所示。 在我们的框架中，$|d_e|$的实验约束比分支比$\mathcal{B}(\mu \to e \gamma)$的约束要严格得多。 假设我们模型参数$\delta_{\alpha}$对于电子的相位为一阶量，我们估计了$\mathcal{B}(\mu \to e \gamma)$的上限，该上限最多为$10^{-21}-10^{-20}$。 如果模型参数$\delta_{\alpha}$、$\delta_{\beta}$（对于缪子）和$\delta_{\gamma}$（对于陶子）是实数，由于模形式的CP相位由模$\tau$引起，不会对EDM产生贡献，因此轻子EDM消失。 然而，我们可以得到$\mathcal{B}(\mu \to e \gamma)\simeq 10^{-13}$，其中$d_e$不为零，这种情况仅当$\delta_{\alpha}$是实数而其他参数是复数时成立。 然后，$\delta_{\beta}$的虚部可以导致下一个领先项中的$d_e$。 预测的电子电偶极矩低于$10^{-32}$e cm，而$\mathcal{B}(\mu \to e \gamma)$接近实验上限。 预测的$|d_\mu|$上限相当大，约为$10^{-23}$e cm，而$|d_\tau|$的为$ 10^{-22}$e cm。 $\tau\to e\gamma$和$\tau\to \mu\gamma$的分支比也进行了讨论。 显示英文摘要

摘要： We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $(g-2)_{\mu}$, in the framework of the SMEFT with the Stringy Ansatz. We have adopted the successful $\Gamma_2$ modular invariant model by Meloni-Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of $(g-2)_{\mu}$, $\Delta a_{\mu}$ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $\Delta a_e$, the electron EDM $d_e$ and the $\mu\to e \gamma$ decay. We found that the NP contributions to $\Delta a_{e(\mu)}$ are proportional to the lepton masses squared likewise the naive scaling $\Delta a_\ell \propto m^2_\ell$. The experimental constraint of $|d_e|$ is much tight compared with the one from the branching ratio $\mathcal{B}(\mu \to e \gamma)$ in our framework. Supposing the phase of our model parameter $\delta_{\alpha}$ for the electron to be of order one, we have estimated the upper-bound of $\mathcal{B}(\mu \to e \gamma)$, which is at most $10^{-21}-10^{-20}$. If the model parameters $\delta_{\alpha}$, $\delta_{\beta}$ (for muon) and $\delta_{\gamma}$ (for tauon) are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus $\tau$ does not contribute to the EDM. However, we can obtain $\mathcal{B}(\mu \to e \gamma)\simeq 10^{-13}$ with non-vanishing $d_e$, that is the case, only $\delta_{\alpha}$ is real while the others are complex. Then, the imaginary part of $\delta_{\beta}$ can lead to $d_e$ in the next-to-leading contribution. The predicted electron EDM is below $10^{-32}$e\,cm, while $\mathcal{B}(\mu \to e \gamma)$ is close to the experimental upper-bound. The predicted upper-bound of $|d_\mu|$ is rather large, around $10^{-23}$e\,cm and of $|d_\tau|$ is $ 10^{-22}$e\,cm. The branching ratios of $\tau\to e\gamma$ and $\tau\to \mu\gamma$ are also discussed.