Leptonic dipole operator with $$Γ _2$$ Γ 2 modular invariance in light of Muon $$(g-2)_μ $$ ( g - 2 ) μ - Takaaki Nomura; Morimitsu Tanimoto; Xing-Yu Wang (European Physical Journal C: Particles and Fields)
Abstract We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $$(g-2)_{μ }$$ ( g - 2 ) μ in the leptonic dipole operator. We have adopted the successful $$Γ _2$$ Γ 2 modular invariant model by Meloni–Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of $$(g-2)_{μ }$$ ( g - 2 ) μ , $$Δ a_{μ }$$ Δ a μ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $$Δ a_e$$ Δ a e , the electron EDM $$d_e$$ d e and the $$μ → e γ $$ μ → e γ decay. We found that the NP contributions to $$Δ a_{e(μ )}$$ Δ a e ( μ ) are proportional to the lepton masses squared likewise the naive scaling $$Δ a_\ell \propto m^2_\ell $$ 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 $$\mathcal {B} (μ → e γ )$$ B ( μ → e γ ) in our framework. Supposing the phase of our model parameter $$δ _{α }$$ δ α for the electron to be of order one, we have estimated the upper-bound of $$\mathcal {B}(μ → e γ )$$ 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 $$τ $$ τ does not contribute to the EDM. However, we can obtain $$\mathcal {B} (μ → e γ )\simeq 10^{-13}$$ 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 $$\mathcal {B} (μ → e γ )$$ B ( μ → e γ ) is close to the experimental upper-bound. The branching ratios of $$τ → eγ $$ τ → e γ and $$τ → μ γ $$ τ → μ γ are also discussed.