Perturbative QCD predictions in fixed order for cross section ratios

In the standard approach, predictions of perturbative Quantum Chromodynamics for ratios of cross sections are computed as the ratio of fixed-order predictions for the numerator and the denominator. Beyond the lowest order in the perturbative expansion, the result does, however, not correspond to a fixed-order prediction for the ratio. This article describes how exact fixed-order results for ratios of arbitrary cross sections can be obtained. The general method for computations in any order of the perturbative expansion is derived, and results for next-to-leading order and next-to-next-to-leading order calculations are given. The approach is applied to theory predictions for various multi-jet cross section ratios measured at hadron colliders. The two methods are compared with each other and with the experimental data. Recommendations are made how to obtain improved theory predictions with more realistic uncertainty estimates.