Top-quark loop corrections in Z+jet and Z+2 jet production

The sophistication of current predictions for $Z+$jet production at hadron colliders necessitates a re-evaluation of any approximations inherent in the theoretical calculations. In this paper we address one such issue, the inclusion of mass effects in top-quark loops. We ameliorate an existing calculation of $Z+1$~jet and $Z+2$~jet production by presenting exact analytic formulae for amplitudes containing top-quark loops that enter at next-to-leading order in QCD. Although approximations based on an expansion in powers of $1\mathrm{/}{m}_t^2$ can lead to poor high-energy behavior, an exact treatment of top-quark loops demonstrates that their effect is small and has limited phenomenological interest.