Renormalization group invariance and optimal QCD renormalization scale-setting: a key issues review

Abstract

A valid prediction for a physical observable from quantum field theory should be independent of the choice of renormalization scheme--this is the primary requirement of renormalization group invariance (RGI). Satisfying scheme invariance is a challenging problem for perturbative QCD (pQCD), since a truncated perturbation series does not automatically satisfy the requirements of the renormalization group. In a previous review, we provided a general introduction to the various scale setting approaches suggested in the literature. As a step forward, in the present review, we present a discussion in depth of two well-established scale-setting methods based on RGI. One is the 'principle of maximum conformality' (PMC) in which the terms associated with the β-function are absorbed into the scale of the running coupling at each perturbative order; its predictions are scheme and scale independent at every finite order. The other approach is the 'principle of minimum sensitivity' (PMS), which is based on local RGI; the PMS approach determines the optimal renormalization scale by requiring the slope of the approximant of an observable to vanish. In this paper, we present a detailed comparison of the PMC and PMS procedures by analyzing two physical observables R(e+e-) and [Formula: see text] up to four-loop order in pQCD. At the four-loop level, the PMC and PMS predictions for both observables agree within small errors with those of conventional scale setting assuming a physically-motivated scale, and each prediction shows small scale dependences. However, the convergence of the pQCD series at high orders, behaves quite differently: the PMC displays the best pQCD convergence since it eliminates divergent renormalon terms; in contrast, the convergence of the PMS prediction is questionable, often even worse than the conventional prediction based on an arbitrary guess for the renormalization scale. PMC predictions also have the property that any residual dependence on the choice of initial scale is highly suppressed even for low-order predictions. Thus the PMC, based on the standard RGI, has a rigorous foundation; it eliminates an unnecessary systematic error for high precision pQCD predictions and can be widely applied to virtually all high-energy hadronic processes, including multi-scale problems.

Setting the renormalization scale in perturbative QCD: Comparisons of the principle of maximum conformality with the sequential extended Brodsky-Lepage-Mackenzie approach

“…As mentioned in the introduction, RG-invariance is broken at fixed order, leading to well-known residual renormalization scale-and scheme-ambiguities. If one requires a fixed-order prediction to satisfy RG-invariance, as suggested by Stevenson [49][50][51] (called local RGinvariance [21]), one can derive an "optimal scale" and even an "optimal scheme" of a process by using the extended RG-equations. This is the method of the "Principal of Minimum Sensitivity (PMS).…”

“…We adopt the conservative procedure suggested by Ref. [21] to predict the "unknown" highorder pQCD corrections; i.e., we identify the perturbative uncertainty with the last known order. Thus the "unknown" high-order pQCD correction is taken as ±|C n a n+1 | MAX for R n , where |C n a n+1 | is calculated by varying µ ∈ [Q/2, 2Q], and the symbol "MAX" stands for the maximum |C n a n+1 | within this scale region.…”

“…Both of these methods satisfy RG-invariance, but they resum the pQCD series in different steps. We have shown that PMC-I and PMC-II are numerically equivalent by comparing several high-energy processes up to the four-loop level [18,21]. A systematic demonstration of this equivalence will appear soon [22].…”

Application of the principle of maximum conformality to the top-quark charge asymmetry at the LHC

“…Renormalization group invariance implies that the prediction for a physical observable cannot depend on the choice of the initial renormalization scale [21][22][23][24][25] or the choice of the renormalization scheme. The Principle of Maximum Conformality (PMC) provides a systematic and unambiguous way to set the renormalization scale and to eliminate the renormalization scheme and scale uncertainty for fixed-order pQCD predictions [26][27][28][29][30][31][32].…”

General properties on applying the principle of minimum sensitivity to high-order perturbative QCD predictions

“…[11,12]. It is noted that the PMS satisfies local RG-invariance [13], which provides a practical approach to systematically fix the optimal scheme and scale for high-energy process. It has been noted that after applying the PMS, the pQCD prediction does show a fast steady behavior over the scheme and scale changes.…”