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

Abstract

Abstract We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $$(g-2)_{\mu }$$ ( g - 2 ) μ in the leptonic dipole operator. We have adopted the successful $$\Gamma _2$$ Γ 2 modular invariant model by Meloni–Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of $$(g-2)_{\mu }$$ ( g - 2 ) μ , $$\Delta a_{\mu }$$ Δ a μ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $$\Delta a_e$$ Δ a e , the electron EDM $$d_e$$ d e and the $$\mu \rightarrow e \gamma $$ μ → e γ decay. We found that the NP contributions to $$\Delta a_{e(\mu )}$$ Δ a e ( μ ) are proportional to the lepton masses squared likewise the naive scaling $$\Delta a_\ell \propto m^2_\ell $$ Δ a ℓ ∝ m ℓ 2 . The experimental constraint of $$|d_e|$$ | d e | is much tight compared with the one from the branching ratio $$\mathcal {B} (\mu \rightarrow e \gamma )$$ B ( μ → e γ ) in our framework. Supposing the phase of our model parameter $$\delta _{\alpha }$$ δ α for the electron to be of order one, we have estimated the upper-bound of $$\mathcal {B}(\mu \rightarrow e \gamma )$$ B ( μ → e γ ) , which is at most $$10^{-21}-10^{-20}$$ 10 - 21 - 10 - 20 . If some model parameters are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus $$\tau $$ τ does not contribute to the EDM. However, we can obtain $$\mathcal {B} (\mu \rightarrow e \gamma )\simeq 10^{-13}$$ B ( μ → e γ ) ≃ 10 - 13 with non-vanishing $$d_e$$ d e in a specific case. The imaginary part of a parameter can lead to $$d_e$$ d e in the next-to-leading contribution. The predicted electron EDM is below $$10^{-32}$$ 10 - 32 e cm, while $$\mathcal {B} (\mu \rightarrow e \gamma )$$ B ( μ → e γ ) is close to the experimental upper-bound. The branching ratios of $$\tau \rightarrow e\gamma $$ τ → e γ and $$\tau \rightarrow \mu \gamma $$ τ → μ γ are also discussed.