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

Abstract

In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to 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 (PMS) 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 invariancereflectivity, symmetry, and transitivity. Using the PMC, all non-conformal {β R i }-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 due to unknown higher-order {βi}-terms can be substantially suppressed. The PMC thus eliminates a serious systematic scale error in pQCD predictions, greatly improving the precision of tests of the Standard Model and the sensitivity to new physics at collider and other experiments.

Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the C -scheme coupling

“…When one applies the PMC, all nonconformal terms are systematically eliminated at each finite order. The PMC satisfies the self-consistency conditions of the renormalization group, such as reflectivity, symmetry, and transitivity [15]. The PMC scales are achieved by applying the RGE recursively.…”

Reanalysis of the Higgs boson decay H → gg up to ${\alpha }_{s}^{6}$-order level using the principle of maximum conformality

“…Residual scale dependence and an estimation of unknown high-order contributions after applying the PMC scale-setting µ r (GeV) PMC scales (GeV) Eqs. (15,16,17,18) There are residual scale dependence due to unknown high-order terms, which however suffer from both the α ssuppression and the exponential suppression. Thus those residual scale dependence are generally small.…”

“…The principle of maximum conformality (PMC) [11][12][13][14] has been suggested for the purpose of eliminating the conventional renormalization scheme-and-scale ambiguities. It has been found that the PMC satisfies all the self-consistency conditions of the renormalization group equation (RGE) or the β-function [15] and the renormalization group invariance, which has been applied for various processes, c.f. the reviews [16][17][18].…”

Higher-twist effects in the direct photon production and the role of infrared renormalons

“…Also, it should be noted that the behavior of the renormalization and factorization scales is widely studied in Refs. [16][17][18][19][20][21]. The infrared renormalon effects in various inclusive and exclusive processes present one of the most important and interesting problems in perturbative QCD [22][23][24][25][26][27].…”