Coherent $\mu-e$ Conversion at Next-to-Leading Order

We analyze next-to-leading order (NLO) corrections and uncertainties for coherent $\mu-e$ conversion . The analysis is general but numerical results focus on ${}^{27}\textrm{Al}$, which will be used in the Mu2E experiment. We obtain a simple expression for the branching ratio in terms of Wilson coefficients associated with possible physics beyond the Standard Model and a set of model-independent parameters determined solely by Standard Model dynamics. For scalar-mediated conversion, we find that NLO two-nucleon contributions can significantly decrease the branching ratio, potentially reducing the rate by as much as 50%. The pion-nucleon $\sigma$-term and quark masses give the dominant sources of parametric uncertainty in this case. For vector-mediated conversion, the impact of NLO contributions is considerably less severe, while the present theoretical uncertainties are comparable to parametric uncertainties.