Lepton flavor violation, lepton $(g-2)_{\mu,\,e}$ and electron EDM in the modular symmetry

We study the lepton flavor violation (LFV), the leptonic magnetic moments $(g-2)_{\mu,\,e} $ and the electric dipole moment (EDM) of the electron in the Standard-Model Effective Field Theory with the $\Gamma_N$ modular flavor symmetry. We employ the stringy Ansatz on coupling structure that 4-point couplings of matter fields are written by a product of 3-point couplings of matter fields. We take the level 3 finite modular group, $\Gamma_3$ for the flavor symmetry, and discuss the dipole operators at nearby fixed point $\tau=i$, where observed lepton masses and mixing angles are well reproduced. Suppose the anomaly of the anomalous magnetic moment of the muon to be evidence of the new physics (NP), we have related it with $(g-2)_{e}$, LFV decays, and the electron EDM. It is found that the NP contribution to $(g-2)_{e}$ is proportional to the lepton masses squared likewise the naive scaling. We also discuss the correlations among the LFV processes $\mu\to e\gamma$, $\tau\to \mu\gamma$ and $\tau\to e\gamma$, which are testable in the future. The electron EDM requires the tiny imaginary part of the relevant Wilson coefficient in the basis of real positive charged lepton masses, which is related to the $\mu\to e\gamma$ transition in our framework.