Leptonic dipole operator with Γ₂ 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)μ, in the framework of the SMEFT with the Stringy Ansatz. We have adopted the successful Γ₂ modular invariant model by Meloni-Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of (g-2)μ, Δ aμ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron Δ aₑ, the electron EDM dₑ and the μ→ e γ decay. We found that the NP contributions to Δ ae(μ) are proportional to the lepton masses squared likewise the naive scaling Δ a_ ∝ m²_. The experimental constraint of |dₑ| is much tight compared with the one from the branching ratio B(μ → e γ) in our framework. Supposing the phase of our model parameter δα for the electron to be of order one, we have estimated the upper-bound of B(μ → e γ), which is at most 10⁻²¹-10⁻²⁰. If the model parameters δα, δβ (for muon) and δγ (for tauon) are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus τ does not contribute to the EDM. However, we can obtain B(μ → e γ) 10⁻¹³ with non-vanishing dₑ, that is the case, only δα is real while the others are complex. Then, the imaginary part of δβ can lead to dₑ in the next-to-leading contribution. The predicted electron EDM is below 10⁻³²e\,cm, while B(μ → e γ) is close to the experimental upper-bound. The predicted upper-bound of |d_μ| is rather large, around 10⁻²³e\,cm and of |d_τ| is 10⁻²²e\,cm. The branching ratios of τ→ eγ and τ→ μγ are also discussed.