Leptonic dipole operator with \(\Gamma _2\) modular invariance in light of Muon \((g-2)_\mu \)

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 leptonic dipole operator. 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 \(\mu \rightarrow e \gamma \) 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 \(\mathcal {B} (\mu \rightarrow e \gamma )\) 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 \(\mathcal {B}(\mu \rightarrow e \gamma )\), which is at most \(10^{-21}-10^{-20}\). If some model parameters 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 \(\mathcal {B} (\mu \rightarrow e \gamma )\simeq 10^{-13}\) with non-vanishing \(d_e\) in a specific case. The imaginary part of a parameter can lead to \(d_e\) in the next-to-leading contribution. The predicted electron EDM is below \(10^{-32}\)e cm, while \(\mathcal {B} (\mu \rightarrow e \gamma )\) is close to the experimental upper-bound. The branching ratios of \(\tau \rightarrow e\gamma \) and \(\tau \rightarrow \mu \gamma \) are also discussed.