Elimination of QCD Renormalization Scale and Scheme Ambiguities

The setting of the renormalization scale ($\mu_r$) in the perturbative QCD (pQCD) is one of the crucial problems for achieving precise fixed-order pQCD predictions. The conventional prescription is to take its value as the typical momentum transfer $Q$ in a given process, and theoretical uncertainties are then evaluated by varying it over an arbitrary range. The conventional scale-setting procedure introduces arbitrary scheme-and-scale ambiguities in fixed-order pQCD predictions. The principle of maximum conformality (PMC) provides a systematic way to eliminate the renormalization scheme-and-scale ambiguities. The PMC method has rigorous theoretical foundations; it satisfies the renormalization group invariance (RGI) and all of the self-consistency conditions derived from the renormalization group. The PMC has now been successfully applied to many physical processes. In this paper, we summarize recent PMC applications, including event shape observables and heavy quark pair production near the threshold region in $e^+e^-$ annihilation and top-quark decay at hadronic colliders. In addition, estimating the contributions related to the uncalculated higher-order terms is also summarized. These results show that the major theoretical uncertainties caused by different choices of $\mu_r$ are eliminated, and the improved pQCD predictions are thus obtained, demonstrating the generality and applicability of the PMC.


I. INTRODUCTION
Quantum Chromodynamics (QCD) is the non-Abelian gauge field theory that describes the strong interactions of quarks and gluons. 50 years ago, the asymptotic freedom property of QCD was proposed by Politzer, Gross and Wilczek [1,2]. Due to the asymptotic freedom property, the strong interaction whose magnitude can be characterized by the strong coupling α s becomes small at very short distances, allowing perturbative calculations for processes involving large momentum transfer. The strong coupling α s is scale dependent, which is controlled by Renormalization Group Equation (RGE) via the β function, The terms β 0 , β 1 , · · · are one-loop, two-loop, · · · coefficients, respectively.
In the framework of perturbative QCD (pQCD), the prediction for an observable ρ at the n th -order level can * email:sqwang@cqu.edu.cn † email:sjbth@slac.stanford.edu ‡ email:wuxg@cqu.edu.cn § email:shenjm@hnu.edu.cn ¶ email:ldigiustino@uninsubria.it be expressed as a perturbative series over the strong coupling α s (µ r ), i.e., where p is the power of the coupling constant for the treelevel terms. The scale µ r represents the initial choice of renormalization scale. The coefficients C 1 , C 2 , · · · are one-loop correction, two-loop correction, · · ·, respectively. The pQCD predictions, calculated up to all orders with n → ∞, are independent to any choices of the renormalization scheme and renormalization scale because of Renormalization Group Invariance (RGI). At any finite order, the renormalization scheme and scale dependence of the coupling constant α s (µ r ) and of the perturbative coefficients C i do not naturally cancel. For example, it has been conventional to guess the renormalization scale µ r as the characteristic momentum flow Q of a process so as to minimize large logarithmic corrections and achieve relativistically more convergent series. This treatment breaks the RGI and introduces arbitrary scheme-andscale dependences in pQCD predictions. Conventional scale-setting also has the negative consequence that the resulting pQCD series suffers from a divergent renormalon (α n s β n 0 n!) series [3] characteristic of a nonconformal series at order n. Furthermore, the theoretical error estimated by simply varying µ r over an arbitrary range such as µ r ∈ [Q/2, 2Q] is clearly unreliable, since it is only sensitive to the β-dependent non-conformal terms, not the entire perturbative series. We actually do not know what is the correct range of variation of the renormalization scale in order to have reliable quantitative predictions for the theoretical uncertainties. Moreover, one also cannot judge whether the poor convergence is the intrinsic property of pQCD series, or is due to the improper choice of renormalization scale. Using the guessied scale is also inconsistent with the well-known Gell-Mann-Low (GM-L) method used in QED [4]. In practice, the GM-L method shows that the correct momentum flow, which is independent to the choice of renormalization scale, can be fixed by resumming all the vacuum polarization diagrams. There is thus no ambiguity in setting the renormalization scale in QED. A self-consistent scale-setting method should be adaptable to both QCD and QED. Predictions of non-Abelian QCD theory must agree analytically with the predictions of Abelian QED, including renormalization scale-setting, in the limit of N C → 0 [5]. Thus eliminating those ambiguities and achieving precise pQCD predictions play crucial roles in testing the Standard Model (SM) and in searching of new physics beyond the SM.
The well-known Brodsky-Lepage-Mackenzie (BLM) method have been suggested in Ref. [6], which is improved to all orders as the Principle of Maximum Conformality (PMC) [7][8][9][10][11] method, provides a systematic all-orders way to eliminate the renormalization scheme-and-scale ambiguities. The PMC provides the underlying principle for the BLM and extends its procedures unambiguously to all orders. The PMC method has a rigorous theoretical foundation, satisfying the RGI [12][13][14] and all of the selfconsistency conditions derived from the renormalization group [15]. The PMC scales are obtained by shifting the argument of α s to eliminate all the non-conformal {β i }terms; the resulting perturbative series thus matches the conformal series with β = 0; the PMC scales thus reflect the virtuality of the propagating gluons for the QCD processes. The divergent renormalon contributions are eliminated, and the resulting perturbative convergence is in general greatly improved. The PMC reduces in the Abelian limit to the GM-L method. The resulting PMC scales also determine the correct effective numbers of active flavors n f at each order.
A crucial point is that the resulting scale-fixed predictions for physical observables using the PMC are independent of the choice of renormalization scheme -a key requirement of RGI. Due to uncalculated higher-order contributions, this leads to residual scale dependence for the PMC scale itself (first kind of residual scale dependence). The last terms of the pQCD approximant are unfixed because of its PMC scale cannot be determined (second kind of residual scale dependence) [16].
In year 2017, the PMC single-scale method (PMCs) [17] has been suggested, which is is equivalent to multi-scale method [7][8][9][10][11] in the sense of perturbative theory. The PMC single-scale method effectively replaces the individual PMC scales at each order derived by using the PMC multi-scale method in the sense of a mean value theorem. The PMC single-scale method exactly removes the second kind of residual scale dependence and it can be regarded as the overall effective (physical) momentum flow of the process. The PMC single-scale method also eliminates the renormalization scheme-and-scale ambiguities and satisfies the standard the RGI. In year 2020, we use an additional property of renormalizable SU(N)/U(1) gauge theories [18], "Intrinsic Conformality (iCF)", which underlies the scale invariance of physical observables. It shows that the scale-invariant perturbative series shows the intrinsic perturbative nature of a pQCD observable. In the year 2022, following the idea of iCF, we have suggested a novel single-scale setting approach under the PMC with the purpose of removing the conventional renormalization scheme-and-scale ambiguities [19]. In Ref. [19], it has been demonstrated that the two PMC single-scale setting methods are equivalent to each other.This equivalence indicates that by using the RGE to fix the value of effective coupling is equivalent to require each loop terms satisfy the scale invariance simultaneously, and vice versa. Thus using the RGE provides a rigorous way to resolve conventional scale-setting ambiguities.
Following the PMC multi-scale procedures [7][8][9][10][11], all the RGE-involved non-conformal {β i }-terms in Eq. (3) are systematically eliminated to fix the correct magnitudes of QCD running couplings at each order (their arguments are called as the PMC scales); the resulting pQCD series then matches the corresponding conformal theory with β = 0, leading to scheme-independent prediction. This is the same principle used in QED where all {β i }-terms that derive from the vacuum polarization corrections of the photon propagator are absorbed into the scale of the QED running coupling. As in QED, the PMC scales are physical in the sense that they reflect the virtuality of the gluon propagators at a given order, as well as they set the effective number n f of active flavors. More explicitly, after applying the PMC multi-scale procedures, the pQCD series for the physical observable ρ becomes where Q i=1,2,3,4 are the PMC scales. Due to uncalculated higher-order contributions, there are two kinds of residual scale dependence [16]. The PMC scale itself is a perturbative expansion series in α s , this leads to residual scale dependence for the PMC scale (first kind of residual scale dependence). In addition, the last terms of the pQCD approximant are unfixed because of its magnitude cannot be determined (second kind of residual scale dependence). These residual scale dependencies are distinct from the conventional scale ambiguities and are suppressed due to the perturbative nature of the PMC scale. In order to suppress the residual scale dependence, which also makes the PMC scale-setting procedures simpler and more easily automatized, the PMC single-scale method has been suggested in Ref. [17]. The PMC singlescale method is equivalent to the multi-scale one in the sense of perturbative theory, and it also provides a self-consistent way to achieve precise α s running behavior in both the perturbative and nonperturbative domains [21,22]. It effectively replaces the individual PMC scales at each order derived by using the PMC multi-scale method in the sense of a mean value theorem. After applying the PMC single-scale procedures, the pQCD prediction for the physical observable ρ can be written as The effective PMC scale Q ⋆ is determined by requiring all the RGE-involved non-conformal terms to vanish simultaneously and can be regarded as the overall effective momentum flow of the process. The PMC single-scale method exactly removes the second kind of residual scale dependence. The scale Q ⋆ shows stability and convergence with increasing order in pQCD, and the first kind of residual scale dependence is thus highly suppressed. The PMC single-scale method eliminates the renormalization scheme-and-scale ambiguities and satisfies the standard the RGI [14]. Until now, the PMC approach has been successfully applied to many high energy processes, including the Higgs boson production at the LHC [23], the Higgs boson decays to γγ [24,25], gg and bb [19,[26][27][28][29] processes, the top-quark pair production at the LHC and Tevatron [8,[30][31][32][33][34][35] and decay process [36], the semihard processes based on the BFKL approach [16,[37][38][39], the electron-positron annihilation to hadrons [10,11,13], the hadronic Z 0 boson decays [40,41], the event shapes in electron-positron annihilation [18,[42][43][44][45], the electroweak parameter ρ [46,47], the Υ(1S) leptonic decay [48,49], the charmonium production [50][51][52] and the decay processes [53][54][55][56]. In addition, the PMC provides a possible solution to the B → ππ puzzle [57] and to the γγ * → η c puzzle [58]. In the following, we present some recent PMC applications and a way of estimating unknown contributions from uncalculated higher-order terms by using the PMC pQCD series.

III. APPLICATIONS
A. New analyses of event shape observables in electron-positron annihilation Event shape observables provide ideal platforms for high precsion tests of QCD. The experiments at LEP and at SLAC have measured event shape distributions with high precision, especially those at the Z 0 peak [59][60][61][62][63]. On the theoretical side, the pQCD corrections to event shape observables have been calculated up to the next-to-next-to-leading order (NNLO) [64][65][66][67][68][69][70]. One of the main purposes of improving the precision of theoretical calculations and experimental measurements is to obtain reliable values of α s (see e.g., [71] for a summary from Particle Data Group). Currently, one finds that the main obstacle for achieving highly precise mesaurements of α s from event shape observables are the theoretical uncertainties, especially from the renormalization scale ambiguity.
Comprehensive PMC analyses for event shape observables in electron-positron annihilation and a novel method for the precise determination of the QCD running coupling have given in Refs. [42,43,45]. Interested readers may turn to these literatures for more details. In this paper, we only present the main PMC results for two classic event shapes, e.g. the thrust (T ) [72,73] and the C-parameter (C) [74,75]. The thrust (T ) and C-parameter (C) are defined as where p i denotes the three-momentum of particle i. For the thrust, the unit vector n is varied to define the thrust direction n T by maximizing the sum on the right-hand side. For the C-parameter, θ ij is the angle between p i and p j . Physical range of values are 1/2 ≤ T ≤ 1 for thrust and 0 ≤ C ≤ 1 for C-parameter respectively. In the case of conventional scale-setting, one simply sets the renormalization scale to be the center-of-mass collision energy µ r = √ s. We present the thrust and C-parameter differential distributions using the conventional scale-setting method at √ s = 91.2 GeV in Fig. 1. Figure 1 shows that even up to NNLO QCD corrections, the conventional predictions are plagued by the large scale uncertainty and substantially deviate from the precise experimental data. By varying µ r ∈ [ √ s/2, 2 √ s], the NLO calculation does not overlap with the LO prediction, and the NNLO calculation does not overlap with NLO prediction. Thus, the estimate of uncalculated higher-order terms for event shape observables by vary- In addition, the perturbative series shows slow convergence because of the renormalon problem. Worse, since the renormalization scale is simply set to µ r = √ s, only one value of α s at the scale √ s can be extracted, whose main error source is the choice of the renormalization scale µ r . The PMC scales are determined by absorbing the β terms of the pQCD series into the coupling constant. We present the PMC scales for the thrust and the Cparameter at √ s = 91.2 GeV in Fig. 2. The resulting PMC scales are not a single value, but they monotonously increase with the value of T and C, reflecting the increasing virtuality of the QCD dynamics. The number of active flavors n f changes with the value of T and C according to the PMC scales. It is noted that the quarks and gluons have soft virtuality near the two-jet region. As the argument of the α s approaches the twojet scale-region, the PMC scales are very soft and thus the non-perturbative effects must be taken into account. The dynamics of the PMC scale thus signals the correct physical behavior in the two-jet region. In addition, the PMC scales are independent of the choice of µ r and are very small in the wide kinematic regions compared to the conventional choice µ r = √ s. It is noted that the behavior of the PMC conformal coefficients is quite different from the pQCD terms givenby the conventional scale-setting method. Since the conformal coefficients are renormalization scale-independent, the resulting PMC predictions eliminate the renormalization scale uncertainty. By setting all input parameters to be their central values, we present the thrust and Cparameter distributions using PMC scale-setting method for √ s = 91.2 GeV in Fig. 3. This figure shows that the PMC predictions are increased in wide kinematic regions compared to the conventional predictions and are in excellent agreement with the experimental data over wide intermediate kinematic regions. Since there are large logarithms which spoil the perturbative regime of the QCD near the two-jet and multi-jet regions, the PMC distributions show some deviations in these two regions. The resummation of large logarithms is thus required for the PMC results especially near the two-jet and multi-jet regions. In fact, the resummation of large logarithms has been extensively studied in the literature.
For the extraction of α s , since the renormalization scale is simply set as µ r = √ s when using conventional scale setting, only one value of α s at scale √ s can be extracted. After applying the PMC method, since the PMC scales vary with the value of the event shapes T and C, we can extract α s (Q 2 ) over a wide range of Q 2 using the experimental data at a single energy of √ s. By comparing PMC predictions with measurements at √ s = 91.2 GeV, we present the extracted running coupling α s (Q 2 ) from the thrust and C-parameter distributions in Fig. 4. Figure 4 shows that the extracted α s (Q 2 ) in the ranges 4 < Q < 16 GeV from the thrust and 3 < Q < 11 GeV from the C-parameter are in excellent agreement with the world average evaluated from the world average α s (M 2 Z ) = 0.1179 [71]. Since the PMC method eliminates the renormalization scale uncertainty, the extracted α s (Q 2 ) is not plagued by any uncertainty from the choice of the scale µ r . Thus, PMC scale-setting provides a remarkable way to verify the running of α s (Q 2 ) from event shape observables in electron-positron annihilation measured at a single energy √ s. The differential distributions of event shape observables are afflicted with large logarithms in the two-jet region. The comparison of QCD predictions with experimental data and then extracting the coupling α s are restricted to the region where leading-twist pQCD theory is able to describe the data well. Choosing different regions of the distributions leads to different values of α s . The mean value of event shape observables provides an important complement to the differential distributions and to determinate α s . The mean value of a event shape y is defined as where y 0 is the kinematically allowed upper limit of the y variable, involves an integration over the full phase space.
In the case of conventional scale setting, the predictions for the mean values of T and C are plagued by the renormalization scale uncertainties and substantially deviate from measurements even up to NNLO [76,77], similar to the case of differential distributions. Currently, the most common way is to split mean values into the perturbative and non-perturbative contributions, which has been studied extensively in the literature. However, some artificial parameters and theoretical models are introduced in order to match theoretical predictions with experimental data.
After applying the PMC, we obtain for the mean value of the thrust, and for the mean value of the C-parameter. The PMC scales satisfy µ pmc r ≪ √ s reflecting the small virtuality of the underlying QCD subprocesses. We note that the analysis of Ref. [59] using conventional scale setting leads to an anomalously large value of α s , demonstrating again that the correct description for the mean values requires µ r ≪ √ s. The PMC scales for the differential distributions of the thrust and C-parameter are also very small. The average of the PMC scales µ pmc r for the differential distributions of the thrust and C-parameter are close to the PMC scales µ pmc r | 1−T and µ pmc r | C , respectively. This shows that PMC scale setting is self-consistent with the differential distributions of the event shapes and their mean values.
After using PMC scale setting, the thrust and Cparameter mean values are increased, especially at small √ s. The scale-independent PMC predictions are in excellent agreement with the experimental data over a wide range of center-of-mass energies √ s [43]. Since we can obtain a high degree of consistency between the PMC predictions and the measurements, the QCD coupling α s (Q 2 ) can be extracted with high precision. The extracted QCD coupling α s (Q 2 ) in the MS scheme from the thrust and C-parameter mean values are presented in Fig. 5. This figure shows that the extracted α s (Q 2 ) are mutually compatible and are in excellent agreement with the world average. The extracted α s (Q 2 ) are not plagued by the renormalization scale uncertainty. In addition, unlike the α s extracted from the differential distributions, the α s extracted from the mean values are not afflicted with large logarithmic contributions nor nonperturbative effects.
A highly precise determination of the value of α s (M 2 Z ) fitting the PMC predictions to the measurements is achieved. Finally, we obtain [43] α s (M 2 Z ) = 0.1185 ± 0.0012, from the thrust mean value, and from the C-parameter mean value. Since the dominant scale µ r uncertainty is eliminated and the convergence of pQCD series is greatly improved after using the PMC, the precision of the extracted α s values is largely improved.
B. Heavy quark pair production in e + e − annihilation near the threshold region Heavy fermion pair production in e + e − annihilation is a fundamental process in the SM. Heavy quark pair production in the threshold region is of particular interest due to the presence of singular terms from the QCD Coulomb corrections. Physically, the renormalization scale which reflects the subprocess virtuality becomes very soft in this region. It is conventional to set the renormalization scale to the mass of the heavy fermion µ r = m f . This conventional procedure obviously violates the physical behavior of the QCD corrections and will lead inevitably to unreliable predictions for the production cross sections in the threshold region. The resummation of logarithmically enhanced terms is thus required.
The quark pair production cross section for e + e − → γ * → QQ at the two-loop level can be written as where a s (µ r ) = α s (µ r )/π, µ r is the renormalization scale. The LO cross section is where α e is the fine structure constant, N c is the number of colors and e Q is the Q quark electric charge. The quark velocity v is v = 1 − 4 m 2 Q /s, where s is the center-ofmass energy squared and m Q is the mass of the quark Q.
The terms δ (2) A , δ (2) L and δ (2) H are the same in either Abelian or non-Abelian theories; the term δ (2) N A only arises in the non-Abelian theory. This process provides the opportunity to explore rigorously the scale-setting method in the non-Abelian and Abelian theories.
The cross section given in Eq.(13) is further divided into the n f -dependent and n f -independent parts, i.e., The coefficients δ v and δ (2) v 2 are for the Coulomb corrections. These coefficients in the MS scheme are calculated in Refs. [79][80][81]. The Coulomb correction plays an important role in the threshold region; it is proportional to powers of (π/v). Thus the renormalization scale is relatively soft in this region. In fact, the PMC scales must be determined separately for the non-Coulomb and Coulomb corrections [8,82]. When the quark velocity v → 0, the Coulomb correction dominates the contribution for the production cross section.
After absorbing the non-conformal term β 0 = 11/3 C A − 4/3 T R n f into the coupling constant using the PMC, we obtain The PMC scales Q i can be written as and the coefficients δ where, i = h and v stand for the non-Coulomb and Coulomb corrections, respectively. The conformal coefficients are independent of the renormalization scale µ r . At the present two-loop level, the PMC scales are also independent of the renormalization scale µ r . Thus, the resulting cross section in Eq. (17) eliminates the renormalization scale uncertainty. Taking C A = 3, C F = 4/3 and T R = 1/2 for QCD, the PMC scales in the MS scheme are Q h = e (−11/24) m Q for the non-Coulomb correction and Q v = 2 e (−5/6) v m Q for the Coulomb correction. The scale Q h originates from the hard gluon virtual corrections, and thus it is determined for the short-distance process. The scale Q v originates from Coulomb rescattering. As expected, the resulting scale Q h is of the order m Q , whereas the scale Q v is of the order v m Q . The scale Q v depends continuously on the quark velocity v, and it becomes soft for v → 0, yielding the correct physical behavior of the scale and reflecting the virtuality of the QCD dynamics.
Effective charge a V s = α V /π (V-scheme) defined by the interaction potential between two heavy quarks [83][84][85][86][87][88][89], V (Q 2 ) = −4 π 2 C F a V s (Q)/Q 2 , provides a physicallybased alternative to the usual MS scheme. As in the case of QED, when the scale of the coupling a V s is identified with the exchanged momentum, all vacuum polarization corrections are resummed into a V s . By using the relation between a s and a V s at the one-loop level, i.e., we convert the quark pair production cross section from the MS scheme to the V-scheme. The corresponding perturbative coefficients in Eq. (16) in the V-scheme are given in Ref. [90]. After applying PMC scale setting in the V-scheme, we obtain the PMC scales Q h = e (3/8) m Q for the non-Coulomb correction and Q v = 2 v m Q for the Coulomb correction. Again, in the V-scheme, Q h is of order m Q , while Q v is of order v m Q . The scale Q v becomes soft for v → 0, and Q v → 2m Q for v → 1, yielding the correct physical behavior. We note that the PMC scales in the usual MS scheme are different from the scales in the physically-based V-scheme. This difference is due to the convention used in defining the MS scheme. The PMC predictions eliminate the dependence from the renormalization scheme; this is explicitly displayed in the form of "commensurate scale relations" (CSR) [91,92].
v,n f |V n f ) is for conventional scale setting and δ For the Coulomb correction, the behavior of the Coulomb term of the form (π/v) δ (2) v is dramatically changed after using the PMC. More explicitly, by taking m Q = 4.89 GeV for the b quark pair production as an example, we present the Coulomb terms of the form (π/v) δ (2) v in the V-scheme using conventional and PMC scale settings in Fig. 6. When the quark velocity v → 0, the Coulomb term is (π/v)δ (2) v → +∞ due to the presence of the term − ln v/v using conventional scale setting. After applying PMC scale setting, the logarithmic term ln(v) vanishes in the coefficient δ (2) v ; the Coulomb term is (π/v)δ (2) v → −∞ due to the term −(π/v). This dramatically different behavior of the (π/v)δ (2) v between conventional and PMC scale settings near the threshold region should be checked in QED.
In analogy to quark pair production, the lepton pair production cross section for the QED process e + e − → γ * → ll is expanded in the QED coupling constant α e . The cross section can also be divided into the non-Coulomb and Coulomb parts, as in the Eq. (16). The perturbative coefficients for the lepton pair production cross section are given in Refs. [79,93,94].
The one-loop correction coefficients δ h,n f , δ (2) v,n f and δ have the same form in QCD and QED with only some replacements: C A = 3, C F = 4/3 and T R = 1/2 in QCD and C A = 0, C F = 1 and T R = 1 in QED.
By using the PMC, the vacuum polarization correc-tions can be absorbed into the QED running coupling: where m i is the mass of the light virtual lepton, and it is far smaller than the final state lepton mass m l . The resulting PMC scales can be written as where, i = h and v stand for the non-Coulomb and Coulomb corrections, respectively. For the lepton pair production, we obtain the PMC scales Q h = e (3/8) m l for the non-Coulomb correction and Q v = 2 v m l for the Coulomb correction.
Since the scales Q h stem from the hard virtual photons corrections and Q v originates from the Coulomb rescattering, Q h is of order m l and Q v is of order v m l . The scales show the same physical behavior from QCD to QED after using PMC scale setting. It is noted that the PMC scales in QCD with the V-scheme coincide with the scales in QED. This scale self-consistency shows that the PMC method in QCD agrees with the standard Gell-Mann-Low method [4] in QED. The V-scheme provides a natural scheme for the QCD process for the quark pair productions.
v,n f n f ) is for conventional scale setting and δ For the Coulomb correction, by taking m τ = 1.777 GeV for the τ lepton as an example, the Coulomb terms of the form (π/v) δ (2) v using conventional and PMC scale settings are shown in Fig. 7. It is noted that in different from the QCD case, when the quark velocity v → 0, the Coulomb terms are (π/v)δ (2) v → −∞ for both the conventional scale setting and the PMC scale setting. Thus, the behavior of the Coulomb term of the form (π/v) δ (2) v is the same using PMC scale setting for both QCD and QED.
In summary, we have shown that two distinctly different scales are determined for the heavy fermion pair production near the threshold region [90]. The PMC scalesetting method in QCD reduces correctly in the Abelian limit N C → 0 to the Gell-Mann-Low method. We also demonstrate the consistency of PMC scale setting in the QED limit.
C. QCD improved top-quark decay at next-to-next-to-leading order Detailed studies of properties of the top-quark such as its mass, its production and the structure of its couplings to other elementary particles plays a crucial role for understanding the nature of electroweak symmetry breaking and searching for new physics beyond the SM. A detailed study of the top-quark decay is highly desirable. The next-to-next-to-leading order (NNLO) QCD corrections to the total width of the top-quark were calculated in Refs. [95][96][97][98]. In recent years, fully differential calculations to the top-quark decay rate at NNLO have been performed in Refs. [99,100]. Experimentally, the Tevatron and LHC experiments have measured the total width of the top-quark decay using different methods. The Particle Data Group (PDG) have reported the world average: Γ t = 1.42 +0. 19 −0.15 GeV [71]. The top-quark decay precess is almost completely dominated by the t → bW , with the subsequent decays of the W bosons into charged leptons, or into quarks. This precess at NNLO can be written as The decay width at leading order (LO) is given by where w = m 2 W /m 2 t , G F is the Fermi constant, |V tb | denotes Cabibbo-Kobayashi-Maskawa (CKM) matrix element, and m t is the mass of the top-quark. The pQCD coefficients c 1 and c 2 obtained using the conventional and PMC scale settings are given in Ref. [36].
SWhen one assumes the conventional scale-setting method, the renormalization scale of α s is usually set to the top-quark mass µ r = m t , and its uncertainty is estimated by varying the scale over an arbitrary range; e.g., µ r ∈ [m t /2, 2m t ]. In Table I    By fixing the scale µ r = m t , the relative importance of the NLO and NNLO QCD correction terms 1% are given in Ref. [99]. Using the same input parameters, our conventional results agree with those of Ref. [99]. However, Table I shows that the NLO and NNLO QCD correction terms change to δΓ After applying PMC scale setting, Table I shows that the PMC results for the NLO and NNLO QCD correction terms are fixed to δΓ NLO t = −0.1892 GeV and δΓ NNLO t = 0.0207 GeV for any choice of the renormalization scale µ r . The relative importance of the NLO and NNLO QCD correction terms is δΓ NLO t /Γ LO t ∼ −12.8% and δΓ NNLO t /Γ LO t ∼ 1.4%. Due to the absorption of the non-conformal terms, the NLO QCD correction term is greatly increased whereas the NNLO QCD correction term is suppressed compared to the conventional results. The renormalization scale uncertainty of conventional scale setting is eliminated. The NNLO QCD correction term provides a negative value using conventional scale setting; it becomes a positive value after using the PMC.
The determined PMC scale for the top-quark decay is The PMC scale is independent of the renormalization scale µ r , and it is one order of magnitude smaller than the conventional choice µ r = m t , reflecting the small virtuality of the QCD dynamics for the top-quark decay process. In addition, the top-quark decay width at NNLO first decreases and then increases with increasing scale µ r using conventional scale setting, achieving its minimum value at µ r ∼ 23 GeV. If one chooses to replace the conventional choice µ r = m t with the small scale µ r ≪ m t (especially around 23 GeV), the pQCD convergence of the top-quark decay will be greatly improved, as well as the resulting conventional prediction decreases and close to the scale-independent PMC prediction. Thus, the effective momentum flow for the top-quark decay process should be µ r ≪ m t , far lower than the conventionally suggested µ r = m t .  In order to provide a reliable prediction for the topquark decay, we need to take into account other corrections such as the effect of finite bottom quark mass and finite W boson width, as well as electroweak corrections. In Table II we present the PMC results of the top-quark decay widths together with the corrections from the finite bottom quark mass, the finite W boson width and the NLO electroweak corrections for m t = 172.5 and 173.5 GeV. These corrections are taken from Ref. [99]. Since the corrections from the finite bottom quark mass and the finite W boson width provide negative values while the NLO electroweak correction provides a positive value, their contributions cancel out greatly to the top-quark decay.
Finally, we obtain reliable predictions for the top-quark total decay width [36] Γ tot t = 1.3112 ± 0.0016 ± 0.0023 GeV (26) for m t = 172.5 GeV, and Γ tot t = 1.3383 +0.0016 −0.0017 ± 0.0023 GeV (27) for m t = 173.5 GeV. Here, the first error comes from the coupling constant ∆α s (M Z ) = ±0.0009 [71] and the second error is caused by the estimation of uncalculated higher-order terms. The top-quark total decay width depends heavily on the top-quark mass, and thus the theoretical error is dominated by the m t . More explicitly, we show the top-quark total decay width Γ tot t versus the top-quark mass m t in Fig. 8. The most precise experimental measurements [101] is also presented as a comparison. Currently, the experimental measurements have D. An estimate of the contributions from uncalculated higher-order terms At present, remarkable progresses have been achieved in doing higher-order calculations in perturbation theory. However due to the complexity of loop calculations, most of perturbatively calculable high-energy observables have only been calculated at lower-orders such as NLO, NNLO and etc. Thus it is important to have a way to estimate the possible contributions from the uncalculated higherorder (UHO) terms such that to improve the predictive power of perturbative theory.
It has been conventional to take µ r as the typical momentum flow Q of the process to obtain the central value of the pQCD series and to then vary µ r within a certain range such as [Q/2, 2Q] as a measure of a combined effect of scale uncertainties and the contributions from the UHO terms. The shortcomings of this treatment are apparent: 1) It's effectiveness heavily depends on the convergence of series which however usually will be diluted by the divergent renormalon terms; 2) Each term in the perturbative series is highly scale-dependent, and the resulting prediction does not satisfy the requirement of RGI; 3) One only partly obtains the information of {β i }-dependent UHO-terms which control the running of α s and no information on the contributions from the conformal {β i }-independent terms. For the more convergent and scale-invariant PMC series, it is expected that a much better prediction of the UHO contributions can be achieved. For the purpose, we need to estimate the magnitude of the UHO-terms in the perturbative series of the pQCD approximant. We also need to know the magnitude of the UHO-terms in the perturbative series of the PMC scale in order to to have an estimatie of the first kind of residual scale dependence.
In this section, we will briefly review two representative approaches to estimate the magnitude of the UHO terms for the perturbative series by using the known partial sum of the conventional and PMC series, respectively. The first approach is to directly predict the magnitude of the UHO coefficient by using a fractional generating function whose parameters can be fixed by matching to the known finite-order series, which is usually called as the Padé approximation approach (PAA) [102][103][104]. The second approach is to quantify the UHO's contribution in terms of a probability distribution whose representative treatment is to use the Bayes' theorem, which is called as the Bayesian-based approach (B.A.) [105][106][107][108].

Estimate of UHO contributions using the Padé approximation approach
The PAA provides a systematic procedure for promoting a finite Taylor series to an analytic function. The PAA offers a feasible conjecture that yields the unknown (n + 1) th -order terms from the given n th -order perturbative series, and a [N/M ]-type fractional generating function ρ c i x i is constructed as [102][103][104] where the parameter M ≥ 1 and N +M = n. The known coefficients c i(≤n) determine the parameters d i∈[0,N ] and e j∈ [1,M] , which inversely predicts a reasonable value for the next uncalculated coefficient c n+1 . For n = 4, it has been observed that the diagonal [2/2]-type Padé series is preferable for estimating the unknown contributions from the conventional pQCD series [109,110]; while the [0/4]-type one is preferable for the PMC series [111], which makes the PAA geometric series be self-consistent with the GM-L prediction [4].

Estimate of UHO contributions using the Bayesian-based approach
The B.A. quantifies the contributions of the UHOterms in terms of the probability distribution, in which the Bayes' theorem is applied to iteratively update the probability as new information becomes available. A detailed introduction of the B.A. and its combination with the PMC approach is given in Ref. [112], so we will only present the main results here, and the interesting readers may turn to Ref. [112] for all the B.A. formulas.
Using B.A., the conditional probability density function (p.d.f.) for a generic (uncalculated) coefficient c n (n > k) of any possible perturbative series ρ k = k i=1 c i α i s with given coefficients {c 1 , c 2 , . . . , c k } is given by, wherec (k) = max{|c 1 |, |c 2 |, · · · , |c k |}. Eq.(29) provides a symmetric probability distribution for negative and positive c n , predicts a uniform probability density in the interval [−c (k) ,c (k) ] and decreases monotonically fromc (k) to infinity. The knowledge of probability density f c (c n |c 1 , c 2 , . . . , c k ) allows one to calculate the degree-of-believe (DoB) that the value of c n belongs to some credible interval (CI). The symmetric smallest CI of fixed p% DoB for c n is denoted by [−c We adopt the interval [−c n α n s ] with p% = 95.5% 1 as the final estimation for any UHO term δ n = c n α n s . As an example, we consider e + e − annihilation ratio R e + e − = σ(e + e − → hadrons)/σ(e + e − → µ + µ − ). We consider the QCD correction of R e + e − , denoted by R(Q). R e + e − (Q) = 3 q e 2 q [1 + R(Q)]. The probability density distributions for R(Q = 31.6 GeV) with different states of knowledge predicted by PMCs and B.A. is presented in Fig. 9. The four lines in each figure correspond to different degrees of knowledge: given LO (dotted), given NLO (dotdashed), given N 2 LO (solid) and given N 3 LO (dashed). The figure illustrates the characteristics of the posterior distribution: a symmetric plateau with two suppressed tails. The posterior distribution given by the Bayesian approach depends on the prior distribution, and as more and more loop terms become known, the probability is updated with less and less dependence on the prior; i.e., the probability density becomes increasingly concentrated (the plateau becomes narrower and narrower and the tail becomes shorter and shorter) as more and more loop terms for the distribution are determined.
As a final remark, because the known coefficients of the conventional pQCD series are scale-dependent at every orders, the PAA and B.A. can only be applied after one On the other hand, the PMC conformal series is scaleindependent, which then provides a more reliable basis for obtaining constraints on the predictions for the UHO contributions. Then the total uncertainty of a pQCD approximant due to the UHO-terms can be treated as the squared average of the predicted conventional scale dependence (or the first kind of residual scale dependence) and the predicted magnitude of the UHO-terms in the perturbative series of the pQCD approximant.

IV. SUMMARY
The setting of the renormalization scale in QCD coupling α s is one of the fundamental problems for pQCD predictions. The conventional scale-setting method introduces inherent scheme-and-scale ambiguities to the pQCD predictions, which becomes one of the most important systematic errors for the pQCD predictions. The PMC method provides a systematic way to eliminate the renormalization scheme-and-scale ambiguities. The PMC method has a rigorous theoretical foundation, satisfying the RGI and all of the self-consistency conditions derived from the renormalization group. The PMC scales are obtained by shifting the argument of α s to eliminate all the non-conformal β-terms; the PMC scales thus reflect the virtuality of the propagating gluons for the QCD processes. The divergent renormalon contributions are eliminated since they are summed in α s , the resulting pQCD convergence is in general greatly improved. The PMC scale-setting method provides the underlying principle for the well-known BLM method, extending the BLM scale-setting procedure unambiguously to all orders. The PMC reduces in the N C → 0 Abelian limit [5] to the GM-L method.
We have provided new analyses for event shape observables in electron-positron annihilation by using the PMC. The resulting PMC scales are not a single value but depend dynamically on the virtuality of the underlying quark and gluon subprocess and thus the specific kinematics of each event. The scale-independent PMC predictions for event shape distributions agree with precise experimental data. Remarkably, the PMC method provides a novel method for the precise determination of the running of α s (Q 2 ) over a wide range of Q 2 from event shape observables measured at a single energy of √ s. The PMC also provides an unambiguous method for determining the scales in multiple-scale processes. It is remarkable that two distinctly different scales are determined for the heavy fermion pair production near the threshold region. One scale is the order of the fermion mass m f , which enters the hard virtual corrections, and the other scale is of order v m f , which enters the Coulomb re-scattering amplitude. Perfect agreement between the Abelian unambiguous Gell-Mann-Low and the PMC scale-setting methods in the limit of zero number of colors is demonstrated in the process of the heavy fermion pair production near the threshold region. We also calculate the top-quark decay process, we obtain the PMC scale Q = 15.5 GeV, reflecting the small virtuality of the QCD dynamics of the top-quark decay process. The convergence of the pQCD series is largely improved for the top-quark decay. Finally, we obtain the top-quark total decay width Γ tot t = 1.3112 +0.0190 −0.0189 GeV. Since the PMC conformal series is scale-independent, it provides a reliable basis for obtaining constraints on the predictions for the UHO contributions. These applications demonstrate the generality and applicability of the PMC. The PMC thus improves the precision tests of the SM and and increases the sensitivity of experiments to new physics beyond the SM.