New Parity-Violating Muonic Forces and the Proton Charge Radius

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 $\sim 100\mathrm{MeV}$ 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.