Self-consistency requirements of the renormalization group for setting the renormalization scale

In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty because of ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG-based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The principle of minimal sensitivity requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The principle of maximum conformality (PMC) satisfies all of the deductions of the RG invariance---reflectivity, symmetry, and transitivity. Using the PMC, all nonconformal βᵢR terms (R stands for an arbitrary renormalization scheme) in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance. Moreover, after PMC scale setting, the residual initial scale dependence at fixed order owing to unknown higher-order βᵢ terms can be substantially suppressed. The PMC thus eliminates a serious systematic scale error in perturbative quantum chromodynamics predictions, greatly improving the precision of tests of the Standard Model and the sensitivity to new physics at collider and other experiments.