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arxiv: 2505.05131 · v2 · submitted 2025-05-08 · ✦ hep-ph

A new method for estimating unknown one-order higher QCD corrections to the perturbative series using the linear regression through the origin

Zhi-Fei Wu , Xing-Gang Wu , Jiang Yan , Xu-Dong Huang , Jian-Ming Shen This is my paper

Pith reviewed 2026-05-22 16:48 UTC · model grok-4.3

classification ✦ hep-ph
keywords perturbative QCDhigher-order correctionsPrinciple of Maximum Conformalitylinear regressionunknown higher-order termsR_tau ratioasymptotic seriesrenormalization scale
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The pith

Linear regression through the origin on scale-invariant perturbative QCD series estimates the next unknown higher-order term.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to estimate unknown higher-order corrections in perturbative QCD calculations. It first applies the Principle of Maximum Conformality to remove renormalization scheme and scale ambiguities from the series. Then it uses linear regression through the origin to determine the form of the asymptotic series and predict the size of the subsequent term. This is tested on the R_tau observable known to four-loop order. If successful it would allow more accurate predictions for high-energy processes without computing full higher-loop diagrams.

Core claim

Using the PMC scheme-and-scale invariant series as the starting point, a novel method of using linear regression through the origin (LRTO) is suggested to fix the asymptotic form of the pQCD series, which subsequently predicts the reasonable magnitude of the one-order higher UHO-terms. As an explicit example, the method is applied to deal with the ratio R_τ, which has been calculated up to four-loop QCD corrections. The results demonstrate that the LRTO method works well, with the scale-invariant and more convergent PMC series exhibiting much better predictive power with stability and reliability than the initial scale-dependent pQCD series.

What carries the argument

The linear regression through the origin (LRTO) applied to the known lower-order terms of the Principle of Maximum Conformality (PMC) improved perturbative series to fix its asymptotic form and extract the next coefficient.

If this is right

  • Reliable estimates of unknown higher-order terms become possible for other QCD observables calculated to finite order.
  • The predictive power and stability increase when the input series is first made scale-invariant via the Principle of Maximum Conformality.
  • Physical predictions such as the value of R_tau can be extended to higher precision without new loop computations.
  • The approach supplies a concrete magnitude for the one-order higher term that can be checked against future exact calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Iterating the regression on successively predicted terms could generate estimates for terms even further beyond the known order.
  • The method might be tested on other observables with existing five-loop results to check consistency across processes.
  • Removing scale dependence first appears to make the underlying asymptotic behavior more transparent for regression fits.
  • If reliable, the technique could complement resummation methods by supplying missing coefficients for improved convergence.

Load-bearing premise

The perturbative QCD series, once made scale-invariant, follows an asymptotic form whose next term can be reliably predicted by performing a linear regression through the origin on the known coefficients.

What would settle it

A direct five-loop calculation of the R_tau ratio that differs significantly from the LRTO prediction would falsify the method's reliability for estimating unknown higher-order terms.

Figures

Figures reproduced from arXiv: 2505.05131 by Jiang Yan, Jian-Ming Shen, Xing-Gang Wu, Xu-Dong Huang, Zhi-Fei Wu.

Figure 1
Figure 1. Figure 1: shows more convergent PMCs series does give more precise fittings: 1) the central values are closer to “exact values” and more quickly tends to steady value with the increment of loop numbers, e.g. the two-loop, [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A comparison of the predicted ln [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The predicted CIs with three DoBs for the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The probability density distributions of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the calculated central values (the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

It is generally believed that the QCD theory is the fundamental theory for strong interactions. Due to the asymptotic freedom at the short distances, after proper factorization, one can predict the value of high-energy physical observable by using the perturbative QCD (pQCD). It has been demonstrated that by recursively using of renormalization group equation with the help of Principle of Maximum Conformality (PMC), one can eliminate conventional renormalization scheme-and-scale ambiguities existed in the initial fixed-order pQCD series. To extend the predictive power of pQCD, we are still facing the problem of how to reliably estimate the contributions from the unknown higher-order (UHO) terms. In this paper, using the PMC scheme-and-scale invariant series as the starting point, we suggest a novel method of using linear regression through the origin (LRTO) to fix the asymptotic form of the pQCD series, which subsequently predicts the reasonable magnitude of the one-order higher UHO-terms. As an explicit example, we apply the method to deal with the ratio $R_\tau$, which has been calculated up to four-loop QCD corrections. Our results show that the LRTO method works well, demonstrating its reliability and significant predictive power for estimating the UHO-terms. Especially, we show that the scale-invariant and more convergent PMC series exhibits a much better predictive power with stability and reliability than the initial scale-dependent pQCD series.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel method that applies linear regression through the origin (LRTO) to the Principle of Maximum Conformality (PMC) improved perturbative series in order to fix an asymptotic form and thereby estimate the magnitude of the next unknown higher-order (UHO) term. The approach is illustrated on the ratio R_τ, which is known through four-loop order; the authors report that the PMC series yields more stable and reliable predictions than the conventional scale-dependent pQCD series.

Significance. If the LRTO procedure on PMC series can be shown to give trustworthy extrapolations, the method would supply a practical, low-cost way to quantify truncation uncertainty in QCD observables. The explicit use of a known four-loop benchmark (R_τ) and the emphasis on the improved convergence properties of the PMC series constitute concrete strengths that can be evaluated directly.

major comments (2)
  1. [Application to R_τ] Application to R_τ (the section presenting numerical results): the claim that LRTO 'works well' on the known four-loop case is not accompanied by an explicit out-of-sample test in which the regression is performed on coefficients up to three loops only and then used to predict the four-loop term (or the five-loop term). Without this demonstration and without reported uncertainties on the fitted slope, the predictive power remains unquantified.
  2. [Method] Method section (description of the asymptotic form and LRTO): the central assumption that the first few known coefficients of the PMC series already obey the linear relation through the origin that is supposed to hold asymptotically is load-bearing for the superiority claim. QCD perturbative series receive their dominant large-order growth from infrared renormalons only at much higher orders; a forced linear fit on n ≤ 4 terms may therefore capture a transient correlation rather than the genuine asymptotic behavior, undermining the extrapolation to the unknown term.
minor comments (2)
  1. [Method] Notation: the symbol for the regression slope and the precise definition of the 'asymptotic form' should be introduced with an equation number and used consistently in all subsequent figures and tables.
  2. [Results] Figure clarity: the plots comparing original and PMC series should include the fitted line, the data points used for the fit, and the predicted next coefficient with its uncertainty band.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments. We address the major comments point by point below, indicating where revisions will be made to improve the manuscript.

read point-by-point responses

  1. Referee: [Application to R_τ] Application to R_τ (the section presenting numerical results): the claim that LRTO 'works well' on the known four-loop case is not accompanied by an explicit out-of-sample test in which the regression is performed on coefficients up to three loops only and then used to predict the four-loop term (or the five-loop term). Without this demonstration and without reported uncertainties on the fitted slope, the predictive power remains unquantified.

    Authors: We agree that an explicit out-of-sample test would strengthen the demonstration of predictive power. In the revised manuscript, we will add a dedicated paragraph in the numerical results section performing the LRTO fit using only the known coefficients up to three-loop order for the PMC-improved R_τ series, then predicting the four-loop coefficient and comparing it directly to the known value. We will also report the uncertainty (e.g., standard error) on the fitted slope parameter from the regression analysis. This will quantify the method's reliability more rigorously. revision: yes

  2. Referee: [Method] Method section (description of the asymptotic form and LRTO): the central assumption that the first few known coefficients of the PMC series already obey the linear relation through the origin that is supposed to hold asymptotically is load-bearing for the superiority claim. QCD perturbative series receive their dominant large-order growth from infrared renormalons only at much higher orders; a forced linear fit on n ≤ 4 terms may therefore capture a transient correlation rather than the genuine asymptotic behavior, undermining the extrapolation to the unknown term.

    Authors: We acknowledge the referee's concern about the validity of assuming asymptotic linear behavior at low orders, given that infrared renormalon contributions typically become dominant at much higher orders. However, the PMC procedure eliminates renormalization-scale and scheme dependence by absorbing all β-function terms into the coupling, yielding a more convergent series in which the linear relation through the origin is observed to hold with greater stability even for the first few terms, as shown by our R_τ results. In the revised manuscript, we will expand the method section with additional discussion and a supporting plot illustrating the improved linearity of the PMC coefficients versus the conventional ones, and we will cite relevant literature on how PMC accelerates the approach to asymptotic behavior. revision: partial

Circularity Check

1 steps flagged

LRTO prediction of UHO term obtained by linear fit to known coefficients of the same series

specific steps
  1. fitted input called prediction [Abstract and method description (LRTO applied to R_tau up to four loops)]
    "using the PMC scheme-and-scale invariant series as the starting point, we suggest a novel method of using linear regression through the origin (LRTO) to fix the asymptotic form of the pQCD series, which subsequently predicts the reasonable magnitude of the one-order higher UHO-terms."

    The asymptotic form is fixed by regressing the known coefficients; the fitted slope is then declared to predict the next coefficient. The estimate is therefore a direct algebraic function of the input terms via the regression, not an independent forecast of the unknown term.

full rationale

The paper proposes LRTO to fix the asymptotic form of the PMC-improved perturbative series and thereby estimate the next unknown coefficient. This procedure fits a linear model (through the origin) directly to the known lower-order coefficients of that series and uses the resulting slope to supply the magnitude of the one-order-higher term. Consequently the claimed prediction is statistically determined by the input coefficients rather than derived from an independent principle or external constraint. The central claim of superior predictive power for the PMC series therefore rests on the assumption that the first few known terms already obey the same linear relation that is supposed to govern the unknown higher orders. No self-citation or ansatz smuggling is required for the reduction; the circularity is internal to the fitting step itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the known perturbative coefficients obey a linear relation through the origin once the series is made scale-invariant; this relation is fitted rather than derived from first principles.

free parameters (1)
  • regression slope
    Determined by linear regression through the origin on the known lower-order coefficients of the PMC series.
axioms (1)
  • domain assumption The perturbative QCD series after PMC improvement admits an asymptotic representation whose next coefficient is captured by a straight line forced through the origin.
    Invoked to convert the known terms into a predictor for the unknown higher-order term.

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Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    Ultraviolet Behavior of Nonabelian Gauge Theories,

    D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Nonabelian Gauge Theories,” Phys. Rev. Lett.30, 1343 (1973). 6 For the case of expansion coefficients exhibiting significant os- cillatory behavior, a suitable preprocessing step to mitigate the oscillations is necessary before applying the LRTO, especially if the couplingα s is not sufficiently small. ...

    work page 1973

  2. [2]

    Reliable Perturbative Results for Strong Interactions?,

    H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev. Lett.30, 1346 (1973)

    work page 1973

  3. [3]

    Feynman, ”inThe Quantum Theory of Fields,” (In- terscience, New York, 1961)

    R.P. Feynman, ”inThe Quantum Theory of Fields,” (In- terscience, New York, 1961)

    work page 1961

  4. [4]

    Generating functionology,

    H.S.Wilf, “Generating functionology,” Am. Math. Mon. 97, 9 (1990)

    work page 1990

  5. [5]

    The Pade approximation and its phys- ical applications,

    J. L. Basdevant, “The Pade approximation and its phys- ical applications,” Fortsch. Phys.20, 283 (1972)

    work page 1972

  6. [6]

    Estimating per- turbative coefficients in quantum field theory using Pade approximants. 2.,

    M. A. Samuel, G. Li and E. Steinfelds, “Estimating per- turbative coefficients in quantum field theory using Pade approximants. 2.,” Phys. Lett. B323, 188 (1994)

    work page 1994

  7. [7]

    Comparison 12 of the Pade approximation method to perturbative QCD calculations,

    M. A. Samuel, J. R. Ellis and M. Karliner, “Comparison 12 of the Pade approximation method to perturbative QCD calculations,” Phys. Rev. Lett.74, 4380 (1995)

    work page 1995

  8. [8]

    Pade approximants, optimal renormal- ization scales, and momentum flow in Feynman dia- grams,

    S. J. Brodsky, J. R. Ellis, E. Gardi, M. Karliner and M. A. Samuel, “Pade approximants, optimal renormal- ization scales, and momentum flow in Feynman dia- grams,” Phys. Rev. D56, 6980 (1997)

    work page 1997

  9. [9]

    Higher-order QCD corrections to hadronicτdecays from Pad´ e approxi- mants,

    D. Boito, P. Masjuan and F. Oliani, “Higher-order QCD corrections to hadronicτdecays from Pad´ e approxi- mants,” JHEP08, 075 (2018)

    work page 2018

  10. [10]

    Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings,

    I. Caprini, “Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings,” Phys. Rev. D100, no.5, 056019 (2019)

    work page 2019

  11. [11]

    alpha(s) and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory,

    M. Beneke and M. Jamin, “alpha(s) and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory,” JHEP09, 044 (2008)

    work page 2008

  12. [12]

    Perturbative ex- pansion ofτhadronic spectral function moments andα s extractions,

    M. Beneke, D. Boito and M. Jamin, “Perturbative ex- pansion ofτhadronic spectral function moments andα s extractions,” JHEP01, 125 (2013)

    work page 2013

  13. [13]

    Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders,

    M. Bonvini, “Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders,” Eur. Phys. J. C80, 989 (2020)

    work page 2020

  14. [14]

    An analysis of Bayesian estimates for missing higher orders in perturbative calculations,

    C. Duhr, A. Huss, A. Mazeliauskas and R. Szafron, “An analysis of Bayesian estimates for missing higher orders in perturbative calculations,” JHEP09, 122 (2021)

    work page 2021

  15. [15]

    Meaningful characterisa- tion of perturbative theoretical uncertainties,

    M. Cacciari and N. Houdeau, “Meaningful characterisa- tion of perturbative theoretical uncertainties,” JHEP09, 039 (2011)

    work page 2011

  16. [16]

    Methods and Applications of Linear Models: Regression and Analysis of Variance. New York: John Wiley

    R. R. Hocking, “Methods and Applications of Linear Models: Regression and Analysis of Variance. New York: John Wiley”. (1996)

    work page 1996

  17. [17]

    J. G. Eisenhauer, “Regression through the Origin. Teach- ing Statistics25, 76 (2003)

    work page 2003

  18. [18]

    Olive, Multiple Linear Regression

    David J. Olive, Multiple Linear Regression. In Linear Regression, Springer International Publishing, charm, 17 (2017)

    work page 2017

  19. [19]

    Predicting the black hole mass and correlations in X-ray reverberating AGN using neural networks,

    P. Chainakun, I. Fongkaew, S. Hancock and A. J. Young, “Predicting the black hole mass and correlations in X-ray reverberating AGN using neural networks,” Mon. Not. Roy. Astron. Soc.513, 648 (2022)

    work page 2022

  20. [20]

    Statistical approach to Higgs boson cou- plings in the standard model effective field theory,

    C. W. Murphy, “Statistical approach to Higgs boson cou- plings in the standard model effective field theory,” Phys. Rev. D97, 015007 (2018)

    work page 2018

  21. [21]

    Normalization of constants in the quanta theory,

    E. C. G. Stueckelberg de Breidenbach and A. Petermann, “Normalization of constants in the quanta theory,” Helv. Phys. Acta26, 499 (1953)

    work page 1953

  22. [23]

    Renormalization Group and the Deep Structure of the Proton,

    A. Peterman, “Renormalization Group and the Deep Structure of the Proton,” Phys. Rept.53, 157 (1979)

    work page 1979

  23. [24]

    Broken scale invariance in scalar field theory,

    C. G. Callan, Jr., “Broken scale invariance in scalar field theory,” Phys. Rev. D2, 1541 (1970)

    work page 1970

  24. [25]

    Small distance behavior in field theory and power counting,

    K. Symanzik, “Small distance behavior in field theory and power counting,” Commun. Math. Phys.18, 227 (1970)

    work page 1970

  25. [26]

    Renormalization Group Invariance and Optimal QCD Renormalization Scale- Setting,

    X. G. Wu, Y. Ma, S. Q. Wang, H. B. Fu, H. H. Ma, S. J. Brodsky and M. Mojaza, “Renormalization Group Invariance and Optimal QCD Renormalization Scale- Setting,” Rept. Prog. Phys.78, 126201 (2015)

    work page 2015

  26. [27]

    The Renor- malization Scale-Setting Problem in QCD,

    X. G. Wu, S. J. Brodsky and M. Mojaza, “The Renor- malization Scale-Setting Problem in QCD,” Prog. Part. Nucl. Phys.72, 44 (2013)

    work page 2013

  27. [28]

    The QCD renormalization group equation and the elimination of fixed-order scheme-and- scale ambiguities using the principle of maximum confor- mality,

    X. G. Wu, J. M. Shen, B. L. Du, X. D. Huang, S. Q. Wang and S. J. Brodsky, “The QCD renormalization group equation and the elimination of fixed-order scheme-and- scale ambiguities using the principle of maximum confor- mality,” Prog. Part. Nucl. Phys.108, 103706 (2019)

    work page 2019

  28. [29]

    High precision tests of QCD without scale or scheme ambiguities: The 40th anniversary of the Brodsky–Lepage–Mackenzie method,

    L. Di Giustino, S. J. Brodsky, P. G. Ratcliffe, X. G. Wu and S. Q. Wang, “High precision tests of QCD without scale or scheme ambiguities: The 40th anniversary of the Brodsky–Lepage–Mackenzie method,” Prog. Part. Nucl. Phys.135, 104092 (2024)

    work page 2024

  29. [30]

    Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops,

    S. J. Brodsky and X. G. Wu, “Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops,” Phys. Rev. D85, 034038 (2012)

    work page 2012

  30. [31]

    Eliminating the Renor- malization Scale Ambiguity for Top-Pair Production Us- ing the Principle of Maximum Conformality,

    S. J. Brodsky and X. G. Wu, “Eliminating the Renor- malization Scale Ambiguity for Top-Pair Production Us- ing the Principle of Maximum Conformality,” Phys. Rev. Lett.109, 042002 (2012)

    work page 2012

  31. [32]

    Setting the Renormal- ization Scale in QCD: The Principle of Maximum Con- formality,

    S. J. Brodsky and L. Di Giustino, “Setting the Renormal- ization Scale in QCD: The Principle of Maximum Con- formality,” Phys. Rev. D86, 085026 (2012)

    work page 2012

  32. [33]

    Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in Perturbative QCD,

    M. Mojaza, S. J. Brodsky and X. G. Wu, “Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in Perturbative QCD,” Phys. Rev. Lett.110, 192001 (2013)

    work page 2013

  33. [34]

    Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations,

    S. J. Brodsky, M. Mojaza and X. G. Wu, “Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations,” Phys. Rev. D89, 014027 (2014)

    work page 2014

  34. [35]

    On the Elimination of Scale Ambiguities in Perturbative Quan- tum Chromodynamics,

    S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, “On the Elimination of Scale Ambiguities in Perturbative Quan- tum Chromodynamics,” Phys. Rev. D28, 228 (1983)

    work page 1983

  35. [36]

    S. J. Brodsky and P. Huet, Aspects of SU(N(c)) gauge theories in the limit of small number of colors, Phys. Lett. B417, 145 (1998)

    work page 1998

  36. [37]

    Quantum electrodynam- ics at small distances,

    M. Gell-Mann and F. E. Low, “Quantum electrodynam- ics at small distances,” Phys. Rev.95, 1300 (1954)

    work page 1954

  37. [38]

    Self-Consistency Re- quirements of the Renormalization Group for Setting the Renormalization Scale,

    S. J. Brodsky and X. G. Wu, “Self-Consistency Re- quirements of the Renormalization Group for Setting the Renormalization Scale,” Phys. Rev. D86, 054018 (2012)

    work page 2012

  38. [39]

    Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of max- imum conformality and theC-scheme coupling,

    X. G. Wu, J. M. Shen, B. L. Du and S. J. Brodsky, “Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of max- imum conformality and theC-scheme coupling,” Phys. Rev. D97, 094030 (2018)

    work page 2018

  39. [40]

    Commensurate scale rela- tions in quantum chromodynamics,

    S. J. Brodsky and H. J. Lu, “Commensurate scale rela- tions in quantum chromodynamics,” Phys. Rev. D51, 3652 (1995)

    work page 1995

  40. [41]

    Generalized Crewther relation and a novel demonstration of the scheme independence of com- mensurate scale relations up to all orders,

    X. D. Huang, X. G. Wu, Q. Yu, X. C. Zheng, J. Zeng and J. M. Shen, “Generalized Crewther relation and a novel demonstration of the scheme independence of com- mensurate scale relations up to all orders,” Chin. Phys. C45, 103104 (2021)

    work page 2021

  41. [42]

    Beneke and V

    M. Beneke and V. M. Braun, Naive nonAbelianization and resummation of fermion bubble chains, Phys. Lett. B348, 513 (1995)

    work page 1995

  42. [43]

    Neubert, Scale setting in QCD and the momentum flow in Feynman diagrams, Phys

    M. Neubert, Scale setting in QCD and the momentum flow in Feynman diagrams, Phys. Rev. D51, 5924 (1995)

    work page 1995

  43. [44]

    Beneke, Renormalons, Phys

    M. Beneke, Renormalons, Phys. Rept.317, 1 (1999)

    work page 1999

  44. [45]

    Ex- tending the Predictive Power of Perturbative QCD,

    B. L. Du, X. G. Wu, J. M. Shen and S. J. Brodsky, “Ex- tending the Predictive Power of Perturbative QCD,” Eur. Phys. J. C79, 182 (2019)

    work page 2019

  45. [46]

    Extending the predic- tive power of perturbative QCD using the principle of maximum conformality and the Bayesian analysis,

    J. M. Shen, Z. J. Zhou, S. Q. Wang, J. Yan, Z. F. Wu, X. G. Wu and S. J. Brodsky, “Extending the predic- tive power of perturbative QCD using the principle of maximum conformality and the Bayesian analysis,” Eur. Phys. J. C83, 326 (2023). 13

    work page 2023

  46. [47]

    Novel All-Orders Single-Scale Approach to QCD Renormaliza- tion Scale-Setting,

    J. M. Shen, X. G. Wu, B. L. Du and S. J. Brodsky, “Novel All-Orders Single-Scale Approach to QCD Renormaliza- tion Scale-Setting,” Phys. Rev. D95, 094006 (2017)

    work page 2017

  47. [48]

    Precise perturbative predictions from fixed-order calculations,

    J. Yan, Z. F. Wu, J. M. Shen and X. G. Wu, “Precise perturbative predictions from fixed-order calculations,” J. Phys. G50, 045001 (2023)

    work page 2023

  48. [49]

    Reanalysis of the BFKL Pomeron at the next-to-leading logarithmic accuracy,

    X. C. Zheng, X. G. Wu, S. Q. Wang, J. M. Shen and Q. L. Zhang, “Reanalysis of the BFKL Pomeron at the next-to-leading logarithmic accuracy,” JHEP10, 117 (2013)

    work page 2013

  49. [50]

    Precise determination of the top-quark on-shell mass via its scale- invariant perturbative relation to the top-quark mass,

    X. D. Huang, X. G. Wu, X. C. Zheng, J. Yan, Z. F. Wu and H. H. Ma, “Precise determination of the top-quark on-shell mass via its scale- invariant perturbative relation to the top-quark mass,” Chin. Phys. C48, 053113 (2024)

    work page 2024

  50. [51]

    Precise Determination of the Bottom-Quark On-Shell Mass Using Its Four-Loop Relation to the MS¯-Scheme Running Mass,

    S. Y. Ma, X. D. Huang, X. C. Zheng and X. G. Wu, “Precise Determination of the Bottom-Quark On-Shell Mass Using Its Four-Loop Relation to the MS¯-Scheme Running Mass,” Chin. Phys. Lett.41, 101201 (2024)

    work page 2024

  51. [52]

    Scale-invariant total decay width Γ(H→b ¯b) using the novel method of characteristic operator,

    J. Yan, X. G. Wu, J. M. Shen, X. D. Huang and Z. F. Wu, “Scale-invariant total decay width Γ(H→b ¯b) using the novel method of characteristic operator,” JHEP04, 184 (2025)

    work page 2025

  52. [53]

    Review of Par- ticle Physics,

    R. L. Workman [Particle Data Group], “Review of Par- ticle Physics,” PTEP2022, 083C01 (2022)

    work page 2022

  53. [54]

    Grigorieva,Series Convergence Theorems and Ap- plications

    E. Grigorieva,Series Convergence Theorems and Ap- plications. In: Methods of Solving Sequence and Series Problems. Birkhauser, Cham, (2016)

    work page 2016

  54. [55]

    An introduction to non-perturbative foun- dations of quantum field theory,

    F. Strocchi, “An introduction to non-perturbative foun- dations of quantum field theory,” Int. Ser. Monogr. Phys. 158, 1 (2013)

    work page 2013

  55. [56]

    On Some possible ex- tensions of the Brodsky-Lepage-MacKenzie approach be- yond the next-to-leading order,

    G. Grunberg and A. L. Kataev, “On Some possible ex- tensions of the Brodsky-Lepage-MacKenzie approach be- yond the next-to-leading order,” Phys. Lett. B279, 352 (1992)

    work page 1992

  56. [57]

    The Generalized Crewther relation in QCD and its experimental consequences,

    S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, “The Generalized Crewther relation in QCD and its experimental consequences,” Phys. Lett. B372, 133 (1996)

    work page 1996

  57. [58]

    Detailed com- parison of renormalization scale-setting procedures based on the principle of maximum conformality,

    X. D. Huang, J. Yan, H. H. Ma, L. Di Giustino, J. M. Shen, X. G. Wu and S. J. Brodsky, “Detailed com- parison of renormalization scale-setting procedures based on the principle of maximum conformality,” Nucl. Phys. B989, 116150 (2023)

    work page 2023

  58. [59]

    Implications of the Principle of Maximum Conformality for the QCD Strong Coupling,

    A. Deur, J. M. Shen, X. G. Wu, S. J. Brodsky and G. F. de Teramond, “Implications of the Principle of Maximum Conformality for the QCD Strong Coupling,” Phys. Lett. B773, 98 (2017)

    work page 2017

  59. [60]

    Novel and self-consistency analysis of the QCD running couplingα s(Q) in both the perturbative and nonpertur- bative domains,

    Q. Yu, H. Zhou, X. D. Huang, J. M. Shen and X. G. Wu, “Novel and self-consistency analysis of the QCD running couplingα s(Q) in both the perturbative and nonpertur- bative domains,” Chin. Phys. Lett.39, 071201 (2022)

    work page 2022

  60. [61]

    Degeneracy Relations in QCD and the Equivalence of Two Systematic All-Orders Methods for Setting the Renormalization Scale,

    H. Y. Bi, X. G. Wu, Y. Ma, H. H. Ma, S. J. Brodsky and M. Mojaza, “Degeneracy Relations in QCD and the Equivalence of Two Systematic All-Orders Methods for Setting the Renormalization Scale,” Phys. Lett. B748, 13 (2015)

    work page 2015

  61. [62]

    The Generalized Scheme-Independent Crewther Relation in QCD,

    J. M. Shen, X. G. Wu, Y. Ma and S. J. Brodsky, “The Generalized Scheme-Independent Crewther Relation in QCD,” Phys. Lett. B770, 494 (2017)

    work page 2017

  62. [63]

    Connections be- tween deep inelastic and annihilation processes at next to next-to-leading order and beyond,

    D. J. Broadhurst and A. L. Kataev, “Connections be- tween deep inelastic and annihilation processes at next to next-to-leading order and beyond,” Phys. Lett. B315, 179 (1993)

    work page 1993

  63. [64]

    Relating inclusive e+ e- annihilation to electroproduction sum rules in quantum chromodynam- ics,

    R. J. Crewther, “Relating inclusive e+ e- annihilation to electroproduction sum rules in quantum chromodynam- ics,” Phys. Lett. B397, 137 (1997)

    work page 1997

  64. [65]

    Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories,

    W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, “Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories,” Phys. Rev. D18, 3998 (1978)

    work page 1978

  65. [66]

    Renormalization Group Improved Pertur- bative QCD,

    G. Grunberg, “Renormalization Group Improved Pertur- bative QCD,” Phys. Lett. B95, 70 (1980)

    work page 1980

  66. [67]

    Renormalization Scheme Independent QCD and QED: The Method of Effective Charges,

    G. Grunberg, “Renormalization Scheme Independent QCD and QED: The Method of Effective Charges,” Phys. Rev. D29, 2315 (1984)

    work page 1984

  67. [68]

    Decays of Heavy Lepton and Intermediate Weak Boson in Quantum Chromodynam- ics,

    C. S. Lam and T. M. Yan, “Decays of Heavy Lepton and Intermediate Weak Boson in Quantum Chromodynam- ics,” Phys. Rev. D16, 703 (1977)

    work page 1977

  68. [69]

    P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Order α4(s) QCD Corrections toZand tau Decays, Phys. Rev. Lett.101, 012002 (2008)

    work page 2008

  69. [70]

    P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Adler Function, Bjorken Sum Rule, and the Crewther Relation to Orderα 4 s in a General Gauge Theory, Phys. Rev. Lett. 104, 132004 (2010)

    work page 2010

  70. [71]

    P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn and J. Rit- tinger, Adler Function, Sum Rules and Crewther Rela- tion of OrderO(α 4 s): the Singlet Case, Phys. Lett. B714, 62 (2012)

    work page 2012

  71. [72]

    P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn and J. Rit- tinger, Vector Correlator in Massless QCD at Order O(α4 s) and the QED beta-function at Five Loop, JHEP 1207, 017 (2012)

    work page 2012

  72. [73]

    K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Run- Dec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun.133, 43 (2000)

    work page 2000

  73. [74]

    Herren and M

    F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun.224, 333 (2018)

    work page 2018