Studying the Proton Radius Puzzle with $\mu$p Elastic Scattering

The Proton Radius Puzzle refers to the disagreement between the proton charge radius as determined from muonic hydrogen and the radius determined from atomic hydrogen level transitions and $ep$ elastic scattering form factor data. The discrepancy of 7$\sigma$ is not yet explained, and though numerous resolutions have been proposed there is no generally accepted resolution to the puzzle. The MUon proton Scattering Experiment (MUSE) Collaboration plans to simultaneously measure $ep$ and $\mu p$ scattering at the Paul Scherrer Institute, testing the interesting possibility that $ep$ and $\mu p$ scattering are different. The experiment will also measure scattering with both polarities of $e$ and $\mu$, directly measuring two-photon exchange effects. In scattering experiments, the proton radius comes from an extrapolation of the form factor slope at low $Q^2$ as $Q^2\rightarrow 0$. We plan to measure in the kinematic region of $Q^2 = 0.002 - 0.07$ GeV$^2$, and determine the relative cross sections to a few tenths of a percent. This should allow the proton radius to be extracted at the level of $\sim$0.01 fm, similar to previous $ep$ measurements. An overview and potential impact of this proposed experiment will be presented.