The 3.6 sigma discrepancy between the predicted and measured values of the anomalous magnetic moment of positive muons can be explained by the existence of a new dark boson Z_mu with a mass in the sub-GeV range, which is coupled predominantly to the
second and third lepton generations through the L_mu - L_tau current . After a discussion of the present phenomenological bounds on the Z_mu coupling, we show that if the Z_mu exists, it could be observed in the reaction mu+Z to mu+Z+Z_mu of a muon scattering off nuclei by looking for an excess of events with large missing muon beam energy in a detector due to the prompt bremsstrahlung Z_mu decay Z_mu to u u into a couple of neutrinos. We describe the experimental technique and the preliminary study of the feasibility for the proposed search. We show that this specific signal allows for a the search for the Z_mu with a sensitivity in the coupling constant alpha_mu > 10^{-11}, which is 3 orders of magnitude higher than the value required to explain the discrepancy. We point out that the availability of high-energy and -intensity muon beams at CERN SPS provides a unique opportunity to either discover or rule out the Z_mu in the proposed search in the near future. The experiment is based on the missing-energy approach developed for the searches for invisible decays of dark photons and (pseudo)scalar mesons at CERN and is complementary to these experiments.
