We introduce a generalization of the conventional renormalization schemes used in dimensional regularization, which illuminates the renormalization scheme and scale ambiguities of perturbative QCD predictions, exposes the general pattern of nonconformal $\{{\beta }_i\}$ 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 perturbative QCD 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.

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 $\beta$-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 $\beta$ 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 ${\alpha }_s^4$; 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 nonconformal 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 nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying $\beta$ terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.

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 $\{{\beta }_i\}$ 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 ${\mu }_R^{\mathrm{PMC}}$ and the resulting finite-order PMC prediction are both to high accuracy independent of the choice of initial renormalization scale ${\mu }_R^{\mathrm{init}}$, 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 $t\overline{t}$-pair production at the Tevatron and LHC colliders. The PMC prediction for the total cross section ${\sigma }_{t\overline{t}}$ 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 ${\mu }_R^{\mathrm{init}}$ equal to $m_t$, $20m_t$, and $\sqrt{s}$. Moreover, after PMC scale setting, we obtain $A_{\mathrm{FB}}^{t\overline{t}}\simeq 12.5\%$, $A_{\mathrm{FB}}^{p\overline{p}}\simeq 8.28\%$ and $A_{\mathrm{FB}}^{t\overline{t}}(M_{t\overline{t}}>450\mathrm{GeV})\simeq 35.0\%$. These predictions have a $1\sigma$ 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.

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of the physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC)/Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all nonconformal $\{{\beta }_i\}$ terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. The PMC/BLM scales can be fixed order-by-order. As a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to next-to-next-to-leading order. An explicit application for determining the scale setting of $R_{e^{+}{e}^{-}}(Q)$ up to four loops is presented. By using the world average ${\alpha }_s^{\overline{\mathrm{MS}}}(M_Z)=0.1184\pm 0.0007$, we obtain the asymptotic scale for the ’t Hooft scheme associated with the $\overline{\mathrm{MS}}$ scheme, ${\Lambda }_{\overline{\mathrm{MS}}}^{\prime tH}={245}_{-10}^{+9}\mathrm{MeV}$, and the asymptotic scale for the conventional $\overline{\mathrm{MS}}$ scheme, ${\Lambda }_{\overline{\mathrm{MS}}}={213}_{-8}^{+19}\mathrm{MeV}$.

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 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.