Discovering leptonic forces using non-conserved currents

Differences in lepton number (i.e., $L _e - L _\mu$, $L _e - L _\tau$, $L_\mu - L _\tau$, or combinations thereof) are not conserved charges in the Standard Model due to the observation of neutrino oscillations. We compute the divergence of the corresponding currents in the case of Majorana or Dirac-type neutrinos and show that, in the high energy limit, the vector interactions map onto those of a light scalar coupled to neutrinos with its coupling fixed by the observed neutrino masses and mixing. This leads to amplitudes with external light vectors that scale inversely with the vector mass. By studying these processes, we set new constraints on $L _i - L _j$ through a combination of semi-leptonic meson decays, invisible neutrino decays, neutrinoless double beta decays, and observations of Big Bang Nucleosynthesis/supernova, which can be much stronger than previous limits for vector masses below an eV. These bounds have important implications on the experimental prospects of detecting $L _i - L _j$ long-range forces.