Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale

摘要

In conventional treatments, predictions from fixed-order perturbative QCDcalculations cannot be fixed with certainty due to ambiguities in the choice ofthe renormalization scale as well as the renormalization scheme. In this paperwe present a general discussion of the constraints of the renormalization group(RG) invariance on the choice of the renormalization scale. We adopt the RGbased equations, which incorporate the scheme parameters, for a generalexposition of RG invariance, since they simultaneously express the invarianceof physical observables under both the variation of the renormalization scaleand the renormalization scheme parameters. We then discuss the self-consistencyrequirements of the RG, such as reflexivity, symmetry, and transitivity, whichmust be satisfied by the scale-setting method. The Principle of MinimalSensitivity (PMS) requires the slope of the approximant of an observable tovanish at the renormalization point. This criterion provides ascheme-independent estimation, but it violates the symmetry and transitivityproperties of the RG and does not reproduce the Gell-Mann-Low scale for QEDobservables. The Principle of Maximum Conformality (PMC) satisfies all of thedeductions of the RG invariance - reflectivity, symmetry, and transitivity.Using the PMC, all non-conformal $\{\beta^{\cal R}_i\}$-terms (${\cal R}$stands for an arbitrary renormalization scheme) in the perturbative expansionseries are summed into the running coupling, and one obtains a unique,scale-fixed, scheme-independent prediction at any finite order. The PMC scalesand the resulting finite-order PMC predictions are both to high accuracyindependent of the choice of initial renormalization scale, consistent with RGinvariance. [...More in the text...]