  The difference in proton radii measured with $\mu p$ atoms and with $ep$
atoms and scattering remains an unexplained puzzle. The PSI MUSE proposal is to
measure $\mu p$ and $e p$ scattering in the same experiment at the same time.
The experiment will determine cross sections, two-photon effects, form factors,
and radii independently for the two reactions, and will allow $\mu p$ and $ep$
results to be compared with reduced systematic uncertainties. These data should
provide the best test of lepton universality in a scattering experiment to
date, about an order of magnitude improvement over previous tests. Measuring
scattering with both particle polarities will allow a test of two-photon
exchange at the sub-percent level, about a factor of four improvement on
uncertainties and over an order of magnitude more data points than previous low
momentum transfer determinations, and similar to the current generation of
higher momentum transfer electron experiments. The experiment has the potential
to demonstrate whether the $\mu p$ and $ep$ interactions are consistent or
different, and whether any difference results from novel physics or two-photon
exchange. The uncertainties are such that if the discrepancy is real it should
be confirmed with $\approx$5$\sigma$ significance, similar to that already
established between the regular and muonic hydrogen Lamb shift.
