  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.
