Target Simulation for the Muon Scattering Experiment (M$\mu$SE) at the Paul Scherrer Institut

In 2010, a novel method of spectroscopic measurements on muonic hydrogen resulted in a 4\% smaller proton radius than previously observed, and at an order of magnitude improvement in precision. This measurement, and a second in 2013, established the so-called "Proton Radius Puzzle". Now, the MUSE collaboration will simultaneously measure, for the first time, electron and muon scattering of both polarities from a liquid hydrogen target to measure the proton radius. To produce a competitive measurement, much work has been done to understand how systematic uncertainties and backgrounds affect the measured lepton-proton elastic scattering cross sections, nuclear form factors, and ultimately the extracted proton radius. To this end, simulation is an invaluable tool for understanding sources of error and guiding experimental design. In this talk, I will review the status of the target simulation for the MUSE experiment and discuss how simulation results have informed the design of the target chamber, liquid hydrogen cells, and background subtraction techniques.