  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.
