In the Standard Model, there are several canonical examples of pure leptonic processes involving the muon, the electron, and the corresponding neutrinos which are connected by the crossing symmetry: i) the decay of muon $\mu \to e{\nu }_{\mu }{\nu }_e$, ii) the inverse muon decay ${\nu }_{\mu }e\to \mu {\nu }_e$, and iii) the annihilation of a muon and an electron into two neutrinos, $\mu e\to {\nu }_{\mu }{\nu }_e$. Although the first two reactions have been observed and measured since long ago, the third process, resulting in the invisible final state, has never been experimentally tested. It may go either directly, or, at low energies, via the annihilation of a muon and an electron from an atomic bound state, called muonium ($M={\mu }^{+}{e}^{-}$). The $M\to {\nu }_e{\overline{\nu }}_{\mu }$ decay is expected to be a very rare process, with the branching fraction predicted to be $\mathrm{Br}(M\to {\nu }_e{\overline{\nu }}_{\mu })=6.6\times {10}^{-12}$ with respect to the ordinary muon decay rate. Using the reported experimental results on precision measurements of the positive muon lifetime by the MuLan Collaboration, we set the first limit $\mathrm{Br}(M\to \mathrm{\text{invisible}})<5.7\times {10}^{-6}$ (90% C.L.), while still leaving a big gap of about six orders of magnitude between this bound and the predictions. To improve substantially the limit, we proposed to perform an experiment dedicated to the sensitive search for the $M\to \mathrm{\text{invisible}}$ decay. A feasibility study of the experimental setup shows that the sensitivity of the search for this decay mode in branching fraction $\mathrm{Br}(M\to \mathrm{\text{invisible}})$ at the level of ${10}^{-12}$ could be achieved. If the proposed search results in a substantially higher branching fraction than predicted, say $\mathrm{Br}(M\to \mathrm{\text{invisible}})\simeq {10}^{-10}$, this would unambiguously indicate the presence of new physics. We point out that such a possibility may occur due to the muonium transition into a hidden sector and consider, as an example, muonium-mirror muonium conversion in the mirror matter model. A result in agreement with the Standard Model prediction would be a theoretically clean check of the pure leptonic bound state annihilation through the charged current weak interactions, which place constraints for further attempts beyond the Standard Model. We believe our work gives strong motivations to perform the proposed experiment on the search for the invisible decay of muonium in the near future.

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