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 [Formula: see text] 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.

We present a detailed analysis on the Bc meson semi-leptonic decays, Bc → ηc(J/ψ)ℓν, up to next-to-leading order (NLO) QCD correction. We adopt the principle of maximum conformality (PMC) to set the renormalization scales for those decays. After applying the PMC scale setting, we determine the optimal renormalization scales for the Bc → ηc(J/ψ) transition form factors (TFFs). Because of the same β0-terms, the optimal PMC scales at the NLO level are the same for all those TFFs, i.e. µ PMC r ≈ 0.8GeV. We adopt a strong coupling model from the massive perturbation theory (MPT) to achieve a reliable pQCD estimation in this low energy region. Furthermore, we adopt a monopole form as an extrapolation for the Bc → ηc(J/ψ) TFFs to all their allowable q 2 region. Then, we predict Γ Bc→ηcℓν (ℓ = e, µ) = (71.53 +11.27 −8.90 ) × 10 −15 GeV, ΓB c→ηc τ ν = (27.14 +5.93 −4.33 ) × 10 −15 GeV, Γ Bc→J/ψℓν (ℓ = e, µ) = (106.31 +18.59 −14.01 ) × 10 −15 GeV, Γ Bc→J/ψτ ν = (28.25 +6.02 −4.35 ) × 10 −15 GeV, where the uncertainties are squared averages of all the mentioned error sources. We show that the present prediction of the production cross section times branching ratio for B + c → J/ψℓ + v relative to that for B + → J/ψK + , i.e. ℜ(J/ψℓ + ν), is in a better agreement with CDF measurements than the previous predictions.

As one of the key components of perturbative QCD theory, it is helpful to find a systematic and reliable way to set the renormalization scale for a high-energy process. The conventional treatment is to take a typical momentum as the renormalization scale, which assigns an arbitrary range and an arbitrary systematic error to pQCD predictions, leading to the well-known renormalization scheme and scale ambiguities. As a practical solution for such scale setting problem, the "Principle of Minimum Sensitivity" (PMS), has been proposed in the literature. The PMS suggests to determine an optimal scale for the pQCD approximant of an observable by requiring its slope over the scheme and scale changes to vanish. In the paper, we present a detailed discussion on general properties of PMS by utilizing three quantities R e + e − , Rτ and Γ(H → bb) up to four-loop QCD corrections. After applying the PMS, the accuracy of pQCD prediction, the pQCD convergence, the pQCD predictive power and etc., have been discussed. Furthermore, we compare PMS with another fundamental scale setting approach, i.e. the Principle of Maximum Conformality (PMC). The PMC is theoretically sound, which follows the renormalization group equation to determine the running behavior of coupling constant and satisfies the standard renormalization group invariance. Our results show that PMS does provide a practical way to set the effective scale for high-energy process, and the PMS prediction agrees with the PMC one by including enough high-order QCD corrections, both of which shall be more accurate than the prediction under the conventional scale setting. However, the PMS pQCD convergence is an accidental, which usually fails to achieve a correct prediction of unknown high-order contributions with next-to-leading order QCD correction only, i.e. it is always far from the "true" values predicted by including more high-order contributions.

Abstract:The complete next-to-next-to-next-to-leading order short-distance and boundstate QCD corrections to Υ(1S) leptonic decay rate Γ(Υ(1S) → + − ) has been finished by Beneke et al. [8]. Based on those improvements, we present a renormalization group (RG) improved pQCD prediction for Γ(Υ(1S) → + − ) by applying the principle of maximum conformality (PMC). The PMC is based on RG-invariance and is designed to solve the pQCD renormalization scheme and scale ambiguities. After applying the PMC, all knowntype of β-terms at all orders, which are controlled by the RG-equation, are resummed to determine optimal renormalization scale for its strong running coupling at each order. We then achieve a more convergent pQCD series, a scheme-independent and more accurate pQCD prediction for Υ(1S) leptonic decay, i.e. Γ Υ(1S)→e + e − | PMC = 1.270 +0.137 −0.187 keV, where the uncertainty is the squared average of the mentioned pQCD errors. This RG-improved pQCD prediction agrees with the experimental measurement within errors.