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, ii) the inverse muon decay, and
iii) the annihilation of a muon and an electron into two neutrinos. 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 Mto u_mu u_e decay is expected to be a very rare process, with the branching fraction predicted to be Br(Mto u_mu u_e) = 6.6 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 Br(M to invisible) < 5.7 10^{-6}, 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 Mto invisible decay. A feasibility study of the experimental setup shows that the sensitivity of the search for this decay mode in branching fraction Br(Mto invisible) at the level of 10^{-12} could be achieved. If the proposed search results in a substantially higher branching fraction than predicted, say Br(M to invisible) < 10^{-10}, this would unambiguously indicate the presence of new physics. We point out that such a possibility may occur due the muonium-mirror muonium conversion in the mirror matter model. A result in agreement with the Standard Model prediction would be a clean check of the pure leptonic bound state annihilation.
