We introduce a generalization of the conventional renormalization schemes used in dimensional regularization, which illuminates the renormalization scheme and scale ambiguities of pQCD predictions, exposes the general pattern of nonconformal {βi} terms, and reveals a special degeneracy of the terms in the perturbative coefficients. It allows us to systematically determine the argument of the running coupling order by order in pQCD in a form which can be readily automatized. The new method satisfies all of the principles of the renormalization group and eliminates an unnecessary source of systematic error.PACS numbers: 12.38. Aw, 12.38.Bx, 11.10.Gh, 11.15.Bt An important goal in high energy physics is to make perturbative QCD (pQCD) predictions as precise as possible, not only to test QCD itself, but also to expose new physics beyond the standard model. In this letter we present a systematic method which determines the argument of the running coupling order by order in pQCD and which can be readily automatized. The resulting predictions for physical processes are independent of theoretical conventions such as the choice of renormalization scheme and the initial choice of renormalization scale. The resulting scales also determine the effective number of quark flavors at each order of perturbation theory. The method can be applied to processes with multiple physical scales and is consistent with QED scale setting in the limit N c → 0. The new method satisfies all of the principles of the renormalization group [1], and it eliminates an unnecessary source of systematic error.The starting point for our analysis is to introduce a generalization of the conventional schemes used in dimensional regularization in which a constant −δ is subtracted in addition to the standard subtraction ln 4π − γ E of the MS-scheme. This amounts to redefining the renormalization scale by an exponential factor; i.e. µ 2 δ = µ 2 MS exp(δ). In particular, the MS-scheme is recovered for δ = ln 4π − γ E . The δ-subtraction defines an infinite set of renormalization schemes which we call * mojaza@cp3-origins.net † sjbth@slac.stanford.edu ‡ wuxg@cqu.edu.cn δ-Renormalization (R δ ) schemes; since physical results cannot depend on the choice of scheme, predictions must be independent of δ. Moreover, since all R δ schemes are connected by scale-displacements, the β-function of the strong QCD coupling constant a = α s /4π is the same in any R δ -scheme:The R δ -scheme exposes the general pattern of nonconformal {β i }-terms, and it reveals a special degeneracy of the terms in the perturbative coefficients which allows us to resum the perturbative series. The resummed series matches the conformal series, which is itself free of any scheme and scale ambiguities as well as being free of a divergent renormalon series. It is the final expression one should use for physical predictions. It also makes it possible to set up an algorithm for automatically computing the conformal series and setting the effective scales for the coupling constant at each pertu...

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.

We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order α 4 s ; i.e. i) the electron-positron annihilation into hadrons, ii) the tau-lepton decay to hadrons, iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the photon field in SU (N ), in accordance with conformality, and with all non-conformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale setting in the Abelian limit is to all orders consistent with QED scale setting. Finally, we demonstrate that the method applies to any renormalization scheme and can be used to derive commensurate scale relations between measurable effective charges, which provide non-trivial tests of QCD to high precision. This work extends BLM scale setting to any perturbative order, with no ambiguities in identifying β-terms in pQCD, demonstrating that BLM scale setting follows from a principle of maximum conformality.

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The conventional scale-setting procedure assigns an arbitrary range and an arbitrary systematic error to fixed-order pQCD predictions. In fact, this ad hoc procedure gives results which depend on the choice of the renormalization scheme, and it is in conflict with the standard scale-setting procedure used in QED. Predictions for physical results should be independent of the choice of scheme or other theoretical conventions. We review current ideas and points of view on how to deal with the renormalization scale ambiguity and show how to obtain renormalization scheme-and scale-independent estimates. We begin by introducing the renormalization group (RG) equation and an extended version, which expresses the invariance of physical observables under both the renormalization scheme and scale-parameter transformations. The RG equation provides a convenient way for estimating the scheme-and scale-dependence of a physical process. We then discuss self-consistency requirements of the RG equations, such as reflexivity, symmetry, and transitivity, which must be satisfied by a scale-setting method. Four typical scale setting methods suggested in the literature, i.e., the Fastest Apparent Convergence (FAC) criterion, the Principle of Minimum Sensitivity (PMS), the Brodsky-Lepage-Mackenzie method (BLM), and the Principle of Maximum Conformality (PMC), are introduced. Basic properties and their applications are discussed. We pay particular attention to the PMC, which satisfies all of the requirements of RG invariance. Using the PMC, all non-conformal terms associated with the β-function in the perturbative series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC provides the principle underlying the BLM method, since it gives the general rule for extending BLM up to any perturbative order; in fact, they are equivalent to each other through the PMC -BLM correspondence principle. Thus, all the features previously observed in the BLM literature are also adaptable to the PMC. The PMC scales and the resulting finite-order PMC predictions are to high accuracy independent of the choice of initial renormalization scale, and thus consistent with RG invariance. The PMC is also consistent with the renormalization scale-setting procedure for QED in the zero-color limit. The use of the PMC thus eliminates a serious systematic scale error in perturbative QCD predictions, greatly improving the precision of empirical tests of the Standard Model and their sensitivity to new physics.

It is conventional to choose a typical momentum transfer of the process as the renormalization scale and take an arbitrary range to estimate the uncertainty in the QCD prediction. However, predictions using this procedure depend on the renormalization scheme, leave a nonconvergent renormalon perturbative series, and moreover, one obtains incorrect results when applied to QED processes. In contrast, if one fixes the renormalization scale using the principle of maximum conformality (PMC), all nonconformal {β(i)} terms 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 scale μ(R)(PMC) and the resulting finite-order PMC prediction are both to high accuracy independent of the choice of initial renormalization scale μ(R)(init), consistent with renormalization group invariance. As an application, we apply the PMC procedure to obtain next-to-next-to-leading-order (NNLO) predictions for the tt-pair production at the Tevatron and LHC colliders. The PMC prediction for the total cross section σ(tt) agrees well with the present Tevatron and LHC data. We also verify that the initial scale independence of the PMC prediction is satisfied to high accuracy at the NNLO level: the total cross section remains almost unchanged even when taking very disparate initial scales μ(R)(init) equal to m(t), 20m(t), and √s. Moreover, after PMC scale setting, we obtain A(FB)(tt)≃12.5%, A(FB)(pp)≃8.28% and A(FB)(tt)(M(tt)>450 GeV)≃35.0%. These predictions have a 1σ deviation from the present CDF and D0 measurements; the large discrepancy of the top quark forward-backward asymmetry between the standard model estimate and the data are, thus, greatly reduced.