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 Al-27, which will be used in the Muon-to-electron conversion (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.