Searching for the dark sector in two-body anti-muon decay with the polarization of monochromatic positrons

The ${\mu }^{+}\to e^{+}X$ decay, where $X$ is a dark sector boson, provides one of the strongest available bounds on the scale of dark sector interactions. The $X$ boson can be an axion or a dark photon. We show that the concurrent determination of the anti-muon and positron polarizations makes possible to distinguish with a confidence level of 99% between the two dark sector portals with as few as 6 observed events in the case of the massless dark photon. Instead, the massive spin-1, dimension 4 dark portal cannot be distinguished from the axion-like case. We also discuss the possibility that the $X$ boson be a massive spin-2 particle.