The QCD renormalization group equation and the elimination of fixed-order scheme-and-scale ambiguities using the principle of maximum conformality

Abstract

The conventional scale setting approach to fixed-order perturbative QCD (pQCD) predictions is based on a guessed renormalization scale, usually taking as the one to eliminate the large log-terms of the pQCD series, together with an arbitrary range to estimate its uncertainty. This ad hoc assignment of the renormalization scale causes the coefficients of the QCD running coupling at each perturbative order to be strongly dependent on the choices of both the renormalization scale and the renormalization scheme, which leads to conventional renormalization scheme-and-scale ambiguities. However, such ambiguities are not necessary, since as a basic requirement of renormalization group invariance (RGI), any physical observable must be independent of the choices of both the renormalization scheme and the renormalization scale. In fact, if one uses the Principle of Maximum Conformality (PMC) to fix the renormalization scale, the coefficients of the pQCD series match the series of conformal theory, and they are thus scheme independent. The PMC predictions also eliminate the divergent renormalon contributions, leading to a better convergence property. It has been found that the elimination of the scale and scheme ambiguities at all orders relies heavily on how precisely we know the analytic form of the QCD running coupling αs. In this review, we summarize the known properties of the QCD running coupling and its recent progresses, especially for its behavior within the asymptotic region. Conventional schemes for defining the QCD running coupling suffer from a complex and scheme-dependent renormalization group equation (RGE), or the β-function, which is usually solved perturbatively at high orders due to the entanglement of the scheme-running and scale-running behaviors. These complications lead to residual scheme dependence even after applying the PMC, which however can be avoided by using a C-scheme coupling αˆs, whose scheme-and-scale running behaviors are governed by the same scheme-independent RGE. As a result, an analytic solution for the running coupling can be achieved at any fixed order. Using the C-scheme coupling, a demonstration that the PMC prediction is scheme-independent to all-orders for any renormalization schemes can be achieved. Given a measurement which sets the magnitude of the QCD running coupling at a specific scale such as MZ, the resulting pQCD predictions, after applying the single-scale PMC, become completely independent of the choice of the renormalization scheme and the initial renormalization scale at any fixed-order, thus satisfying all of the conditions of RGI. An improved pQCD convergence provides an opportunity of using the resummation procedures such as the Padé approximation (PA) approach to predict higher-order terms and thus to increase the precision, reliability and predictive power of pQCD theory. In this review, we also summarize the current progress on the PMC and some of its typical applications, showing to what degree the conventional renormalization scheme-and-scale ambiguities can be eliminated after applying the PMC. We also compare the PA approach for the conventional scale-dependent pQCD series and the PMC scale-independent conformal series. We observe that by using the conformal series, the PA approach can provide a more reliable estimate of the magnitude of the uncalculated terms. And if the conformal series for an observable has been calculated up to nth-order level, then the [N∕M]=[0∕n−1]-type PA series provides an important estimate for the higher-order terms.