The principle of maximum conformality (PMC) has been suggested to eliminate the renormalization scheme and renormalization scale uncertainties, which are unavoidable for the conventional scale setting and are usually important errors for theoretical estimations. In this paper, by applying PMC scale setting, we analyze two important inclusive Standard Model Higgs decay channels, H → bb and H → gg, up to four-loop and three-loop levels, respectively. After PMC scale setting, it is found that the conventional scale uncertainty for these two channels can be eliminated to a high degree. There is small residual initial scale dependence for the Higgs decay widths due to unknown higher-order {β i } terms. Up to four-loop level, we obtain (H → bb) = 2.389 ± 0.073 ± 0.041 MeV and up to threeloop level, we obtain (H → gg) = 0.373 ± 0.030 MeV, where the first error is caused by varying M H = 126 ± 4 GeV and the second error for H → bb is caused by varying the MS-running mass m b (m b ) = 4.18 ± 0.03 GeV. Taking H → bb as an example, we present a comparison of three BLM-based scale-setting approaches, e.g. the PMC-I approach based on the PMC-BLM correspondence, the R δ -scheme and the seBLM approach, all of which are designed to provide effective ways to identify non-conformal {β i }-series at each perturbative order. At four-loop level, all those approaches lead to good pQCD convergence, they have almost the same pQCD series, and their predictions are almost independent on the initial renormalization scale. In this sense, those approaches are equivalent to each other.

The Principle of Maximal Conformality (PMC) provides a rigorous method for eliminating renormalization scheme-and-scale ambiguities for perturbative QCD predictions. The PMC uses the renormalization group equation to fix the β-pattern of each order in an arbitrary pQCD approximant, and it then determines the optimal renormalization scale by absorbing all {βi} terms into the running coupling at each order. The resulting coefficients of the pQCD series match the schemeindependent conformal series with β = 0. As in QED, different renormalization scales appear at each order; we call this the multi-scale approach. In this paper, we present a novel single-scale approach for the PMC, in which a single effective scale is constructed to eliminate all non-conformal β-terms up to a given order simultaneously. The PMC single-scale approach inherits the main features of the multi-scale approach; for example, its predictions are scheme-independent, and the pQCD convergence is greatly improved due to the elimination of divergent renormalon terms. As an application of the single-scale approach, we investigate the e + e − annihilation cross-section ratio R e + e − and the Higgs decay-width Γ(H → bb), including four-loop QCD contributions. The resulting predictions are nearly identical to the multi-scale predictions for both the total and differential contributions. Thus in many cases, the PMC single-scale approach PMC-s, which requires a simpler analysis, could be adopted as a reliable substitution for the PMC multi-scale approach for setting the renormalization scale for high-energy processes, particularly when one does not need detailed information at each order. The elimination of the renormalization scale uncertainty increases the precision of tests of the Standard Model at the LHC.

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 logterms 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-andscale 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 α s . 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 couplingα s , 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 M Z , 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 ...

The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling αs at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this paper, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Padé Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables R e + e − , Rτ , and Γ(H → bb) which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order α p s ) has been determined at order α n s , then the [N/M ] = [0/n − p] Padé series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic e + e − annihilation. Thus, by using the combination of the PMC series and the Padé method, the predictive power of pQCD theory can be greatly improved.