  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$\chi$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 $\mu p$ 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 $\mu p$ and $ep$ cross sections at MUSE as functions of the
assumed value of the proton radius.
