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.

The principle of maximum conformality (PMC) provides a systematic and process-independent method to derive renormalization scheme- and scale-independent fixed-order pQCD predictions. By Brodsky and Wu [Phys. Rev. D 85, 114040 (2012)], we studied the top-quark charge asymmetry at the Tevatron. By applying the PMC, we have shown that the large discrepancies for the top-quark charge asymmetry between the Standard Model estimate and the CDF and D0 data are greatly reduced. In the present paper, with the help of the Bernreuther-Si program, we present a detailed PMC analysis on the top-quark pair production up to next-to-next-to-leading-order level at the LHC. After applying PMC scale setting, the pQCD prediction for the top-quark charge asymmetry at the LHC has very small scale uncertainty; e.g., $A_{\mathrm{C}}{|}_{7\mathrm{TeV};\mathrm{PMC}}=(1.15_{-0.03}^{+0.01})\%$, $A_{\mathrm{C}}{|}_{8\mathrm{TeV};\mathrm{PMC}}=(1.03_{+0.00}^{+0.01})\%$, and $A_{\mathrm{C}}{|}_{14\mathrm{TeV};\mathrm{PMC}}=(0.62_{-0.02}^{+0.00})\%$. The corresponding predictions using conventional scale setting are: $A_{\mathrm{C}}{|}_{7\mathrm{TeV};\text{Conv}}=(1.23_{-0.14}^{+0.14})\%$, $A_{\mathrm{C}}{|}_{8\mathrm{TeV};\text{Conv}}=(1.11_{-0.13}^{+0.17})\%$, and $A_{\mathrm{C}}{|}_{14\mathrm{TeV};\text{Conv}}=(0.67_{-0.05}^{+0.05})\%$. In these predictions, the scale errors are predicted by varying the initial renormalization and factorization scales in the ranges ${\mu }_r^{\text{init}}\in [m_t/2,2m_t]$ and ${\mu }_f\in [m_t/2,2m_t]$. The PMC predictions are also in better agreement with the available ATLAS and CMS data. In addition, we have calculated the top-quark charge asymmetry assuming several typical cuts on the top-pair invariant mass $M_{t\overline{t}}$. For example, assuming $M_{t\overline{t}}>0.5\mathrm{TeV}$ and ${\mu }_f={\mu }_r^{\text{init}}=m_t$, we obtain $A_{\mathrm{C}}{|}_{7\mathrm{TeV};\mathrm{PMC}}=2.67\%$, $A_{\mathrm{C}}{|}_{8\mathrm{TeV};\mathrm{PMC}}=2.39\%$, and $A_{\mathrm{C}}{|}_{14\mathrm{TeV};\mathrm{PMC}}=1.28\%$.

We identify a property of renormalizable SU(N)/U(1) gauge theories, intrinsic conformality (iCF), which underlies the scale invariance of physical observables and leads to a remarkably efficient method to solve the conventional renormalization scale ambiguity at every order in perturbative QCD (pQCD): the ${\mathrm{PMC}}_{\infty }$. This new method reflects the underlying conformal properties displayed by pQCD at next-to-next-to-leading order, eliminates the scheme dependence of pQCD predictions, and is consistent with the general properties of the principle of maximum conformality (PMC). We introduce a new method to identify conformal and $\beta$-terms which can be applied from either a numerical or an analytical calculations. We illustrate the ${\mathrm{PMC}}_{\infty }$ for the thrust and C-parameter distributions in $e^{+}{e}^{-}$ annihilation and then show how to apply this new method to general observables in QCD. We point out how the implementation of the ${\mathrm{PMC}}_{\infty }$ can significantly improve the precision of pQCD predictions; its implementation in a multiloop analysis also simplifies the calculation of higher order corrections in a general renormalizable gauge theory.

We present a detailed analysis on the $B_c$ meson semileptonic decays, $B_c\to {\eta }_c(J/\psi )\ell \nu$, 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 $B_c\to {\eta }_c(J/\psi )$ transition form factors (TFFs). Because of the same ${\beta }_0$-terms, the optimal PMC scales at the NLO level are the same for all those TFFs, i.e., ${\mu }_r^{\mathrm{PMC}}\approx 0.8\mathrm{GeV}$. We adopt a strong coupling model from the massive perturbation theory to achieve a reliable pQCD estimation in this low energy region. Furthermore, we adopt a monopole form as an extrapolation for the $B_c\to {\eta }_c(J/\psi )$ TFFs to all their allowable $q^2$ region. Then, we predict ${\mathrm{\Gamma }}_{B_c\to {\eta }_c\ell \nu }(\ell =e,\mu )=(71.53_{-8.90}^{+11.27})\times 10^{-15}\mathrm{GeV}$, ${\mathrm{\Gamma }}_{B_c\to {\eta }_c\tau \nu }=(27.14_{-4.33}^{+5.93})\times 10^{-15}\mathrm{GeV}$, ${\mathrm{\Gamma }}_{B_c\to J/\psi \ell \nu }(\ell =e,\mu )=(106.31_{-14.01}^{+18.59})\times 10^{-15}\mathrm{GeV}$, ${\mathrm{\Gamma }}_{B_c\to J/\psi \tau \nu }=(28.25_{-4.35}^{+6.02})\times 10^{-15}\mathrm{GeV}$, 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 $B_c^{+}\to J/\psi {\ell }^{+}v$ relative to that for $B^{+}\to J/\psi K^{+}$, i.e., $\Re (J/\psi {\ell }^{+}\nu )$, is in a better agreement with CDF measurements than the previous predictions.