Charged lepton flavor violating decays with a pair of light dark matter and muonium invisible decay

Abstract: In this paper, we initiate the study of lepton flavor violating (LFV) dark matter (DM) interactions, expanding our focus beyond the flavor-conserving DM interactions typically considered in conventional direct and indirect detections. We work on an effective field theory (EFT) framework, focusing on the leading-order local operators of the form, $\bar \ell_j \Gamma \ell_i\,{\tt DM}^2$, where $(ij)=(e\mu, e\tau, \mu\tau)$ and the DM includes the three well-known scenarios: a scalar, a fermion, and a vector. We derive the invariant-mass distribution for the three-body decay $\ell_i \to \ell_j +{\tt DM+DM}$ and demonstrate that it can be used to distinguish between different operator structures and to determine the DM mass. By utilizing current experimental bounds on the charged LFV decays involving neutrinos or single invisible particles, we establish stringent limits on the effective scale associated with each operator. Additionally, for the $e\mu$ flavor combination, we investigate the muon four-body radiative decay ($\mu\to e +{\tt DM+DM}+\gamma$) to complement our probe of such interactions. Finally, we examine muonium invisible decays based on the derived bounds on the effective operators and find that the branching ratios can be significantly enhanced compared to the predictions of the standard model. In particular, any future observation of the para-muonium invisible decay serves as a compelling signature for these flavored DM interactions.