  Positronium and Muonium are purely leptonic atoms and hence free of an
internal sub-structure. This qualifies them as potentially well suited systems
to probe the existence of physics beyond the Standard Model. We hence carry out
a comprehensive study of the sensitivity of current Positronium and Muonium
precision spectroscopy to several new physics scenarios. By taking properly
into account existing experimental and astrophysical probes, we define clear
experimental targets to probe new physics via precise spectroscopy. For
Positronium we find that, in order for the spectroscopy bounds to reach a
sensitivity comparable to the electron gyromagnetic factor, an improvement of
roughly five orders of magnitude from state-of-the-art precision is required,
which would be a challenge based on current technology. More promising is
instead the potential reach of Muonium spectroscopy: in the next few years
experiments like Mu-MASS at PSI will probe new regions of the parameter space
testing the existence of medium/short range (MeV and above) spin-dependent and
spin-independent dark forces between electrons and muons.
