  The recent discrepancy between proton charge radius measurements extracted
from electron-proton versus muon-proton systems is suggestive of a new force
that differentiates between lepton species. We identify a class of models with
gauged right-handed muon number, which contains new vector and scalar force
carriers at the 100 MeV scale or lighter, that is consistent with observations.
Such forces would lead to an enhancement by several orders-of-magnitude of the
parity-violating asymmetries in the scattering of low-energy muons on nuclei.
The relatively large size of such asymmetries, O(10^{-4}), opens up the
possibility for new tests of parity violation in neutral currents with existing
low-energy muon beams.
