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...

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.