Leptonic dipole operator with modular invariance in light of Muon

Takaaki Nomura; Morimitsu Tanimoto; Xing-Yu Wang

doi:10.1140/epjc/s10052-024-13666-w

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2024) 84: 1329
https://doi.org/10.1140/epjc/s10052-024-13666-w

Regular Article - Theoretical Physics

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

Takaaki Nomura¹, Morimitsu Tanimoto² and Xing-Yu Wang¹^a

¹ College of Physics, Sichuan University, 610065, Chengdu, China
² Department of Physics, Niigata University, 950-2181, Niigata, Japan

^a xingyuwang@stu.scu.edu.cn

Received: 14 October 2024
Accepted: 28 November 2024
Published online: 29 December 2024

Abstract

We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, ${(g - 2)}_{μ}$ $(g-2)_{\mu }$ in the leptonic dipole operator. We have adopted the successful $Γ_{2}$ $\Gamma _2$ modular invariant model by Meloni–Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of ${(g - 2)}_{μ}$ $(g-2)_{\mu }$ , $Δ a_{μ}$ $\Delta a_{\mu }$ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $Δ a_{e}$ $\Delta a_e$ , the electron EDM $d_{e}$ $$d_e$$ and the $μ \to e γ$ $\mu \rightarrow e \gamma$ decay. We found that the NP contributions to $Δ a_{e (μ)}$ $\Delta a_{e(\mu )}$ are proportional to the lepton masses squared likewise the naive scaling $Δ a_{ℓ} \propto m_{ℓ}^{2}$ $\Delta a_\ell \propto m^2_\ell$ . The experimental constraint of $| d_{e} |$ $$|d_e|$$ is much tight compared with the one from the branching ratio $B (μ \to e γ)$ $\mathcal {B} (\mu \rightarrow e \gamma )$ in our framework. Supposing the phase of our model parameter $δ_{α}$ $\delta _{\alpha }$ for the electron to be of order one, we have estimated the upper-bound of $B (μ \to e γ)$ $\mathcal {B}(\mu \rightarrow e \gamma )$ , 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 $τ$ $\tau$ does not contribute to the EDM. However, we can obtain $B (μ \to e γ) ≃ 10^{- 13}$ $\mathcal {B} (\mu \rightarrow e \gamma )\simeq 10^{-13}$ 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 $B (μ \to e γ)$ $\mathcal {B} (\mu \rightarrow e \gamma )$ is close to the experimental upper-bound. The branching ratios of $τ \to e γ$ $\tau \rightarrow e\gamma$ and $τ \to μ γ$ $\tau \rightarrow \mu \gamma$ are also discussed.

© The Author(s) 2024

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