Leptonic dipole operator with ₂ modular invariance

We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, (g-2) _, in the framework of the SMEFT with the Stringy Ansatz. We have adopted the successful ₂ modular invariant model by Meloni-Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of (g-2) _, a_ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron aₑ, the electron EDM dₑ and the e decay. We found that the NP contributions to a₄ () are proportional to the lepton masses squared likewise the naive scaling a_ m²_. The experimental constraint of |dₑ| is much tight compared with the one from the branching ratio B (e) in our framework. Supposing the phase of our model parameter _ for the electron to be of order one, we have estimated the upper-bound of B (e), which is at most 10^-21-10^-20. If the model parameters _, _ (for muon) and _ (for tauon) are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus does not contribute to the EDM. However, we can obtain B (e) 10^-13 with non-vanishing dₑ, that is the case, only _ is real while the others are complex. Then, the imaginary part of _ can lead to dₑ in the next-to-leading contribution. The predicted electron EDM is below 10^-32e\, cm, while B (e) is close to the experimental upper-bound. The predicted upper-bound of |d_| is rather large, around 10^-23e\, cm and of |d_| is 10^-22e\, cm. The branching ratios of e and are also discussed.