  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 M\to \nu_\mu \nu_e decay is expected to
be a very rare process, with the branching fraction predicted to be Br(M\to
\nu_\mu\nu_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 M\to invisible decay. A feasibility study of the experimental
setup shows that the sensitivity of the search for this decay mode in branching
fraction Br(M\to 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.
