Distinguishing charged lepton flavor violation scenarios with inelastic $\mu\rightarrow e$ conversion

The Mu2e and COMET experiments are expected to improve existing limits on charged lepton flavor violation (CLFV) by roughly four orders of magnitude. $\mu\rightarrow e$ conversion experiments are typically optimized for electrons produced without nuclear excitation, as this maximizes the electron energy and minimizes backgrounds from the free decay of the muon. Here we argue that Mu2e and COMET will be able to extract additional constraints on CLFV from inelastic $\mu \rightarrow e$ conversion, given the $^{27}$Al target they have chosen and backgrounds they anticipate. We describe CLFV scenarios in which inelastic CLFV can induce measurable distortions in the near-endpoint spectrum of conversion electrons, including cases where certain contributing operators cannot be probed in elastic $\mu \rightarrow e$ conversion. We extend the nonrelativistic EFT treatment of elastic $\mu \rightarrow e$ conversion to include the new nuclear operators needed for the inelastic process, evaluate the associated nuclear response functions, and describe several new-physics scenarios where the inelastic process can provide additional information on CLFV.