The MUSE experiment at Paul Scherrer Institute will perform the first measurement of low-energy muon-proton elastic scattering (muon lab momenta 115-210 MeV) with the aim of determining the proton charge radius. We study the prospects for the proton radius extraction using the theoretical framework of Dispersively Improved Chiral Effective Field Theory (DI$χ$EFT). It connects the proton radii with the finite-$Q^2$ behavior of the form factors through complex analyticity and enables the use of data up to $Q^2 \sim$ 0.1 GeV$^2$ for radius extraction. We quantify the sensitivity of the $μp$ cross section to the proton charge radius, the theoretical uncertainty of the cross section predictions, and the size of two-photon exchange corrections. We find that the optimal kinematics for radius extraction at MUSE is at momenta 210 MeV and $Q^2 \sim$ 0.05-0.08 GeV$^2$. We compare the performance of electron and muon scattering in the same kinematics. As a byproduct, we obtain explicit predictions for the $μp$ and $ep$ cross sections at MUSE as functions of the assumed value of the proton radius.

12 pages, 12 figures