Nuclear-level effective theory of $\mu\rightarrow e$ conversion: Inelastic process

Mu2e and COMET will search for electrons produced via the neutrinoless conversion of stopped muons bound in 1s atomic orbits of $^{27}$Al, improving existing limits on charged lepton flavor violation (CLFV) by roughly four orders of magnitude. Conventionally, $\mu\rightarrow e$ conversion experiments are optimized to detect electrons originating from transitions where the nucleus remains in the ground state, thereby maximizing the energy of the outgoing electron. Clearly, detection of a positive signal in forthcoming experiments would stimulate additional work $-$ including subsequent conversion experiments using complementary nuclear targets $-$ to further constrain the new physics responsible for CLFV. Here we argue that additional information can be extracted without the need for additional experiments, by considering inelastic conversion in $^{27}$Al. Transitions to low-lying nuclear excited states can modify the near-endpoint spectrum of conversion electrons, with the ratio of the elastic and inelastic responses being sensitive to the underlying CLFV operator. We extend the nuclear effective theory of $\mu\rightarrow e$ conversion to the inelastic case, which adds five new response functions to the six that arise for the elastic process. We evaluate these nuclear response functions in $^{27}$Al and calculate the resulting conversion-electron signal, taking into account the resolution anticipated in Mu2e/COMET. We find that $^{27}$Al is an excellent target choice from the perspective of the new information that can be obtained from inelastic $\mu \rightarrow e$ conversion.