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

Abstract

We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $(g-2{)}_{\mu }$ 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 $\mu \to e\gamma$ decay. We found that the NP contributions to $\Delta a_{e(\mu )}$ are proportional to the lepton masses squared likewise the naive scaling $\Delta a_{\ell }\propto m_{\ell }^2$. The experimental constraint of $|d_e|$ is much tight compared with the one from the branching ratio $\mathcal{B}(\mu \to e\gamma )$ 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 $\mathcal{B}(\mu \to e\gamma )$, 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 $\mathcal{B}(\mu \to e\gamma )\simeq 10^{-13}$ 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 $\mathcal{B}(\mu \to e\gamma )$ is close to the experimental upper-bound. The branching ratios of $\tau \to e\gamma$ and $\tau \to \mu \gamma$ are also discussed.

Comments: 37 pages, 12 figures. Version accepted for publication in EPJC