pith. machine review for the scientific record. sign in

arxiv: 2605.07618 · v1 · submitted 2026-05-08 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

New Determinations of the Charm and Bottom Quark Masses Using QCD Quarkonium Sum Rules

Qing Yu , Hua Zhou , Xing-Gang Wu
Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:30 UTC · model grok-4.3

classification ✦ hep-ph
keywords QCD sum rulesheavy quark massesPrinciple of Maximum Conformalityquarkonium correlatorsperturbative QCD correctionsPadé approximationcharm quark massbottom quark mass
0
0 comments X

The pith

Applying the characteristic operator extension of the Principle of Maximum Conformality to QCD quarkonium sum rules eliminates renormalization ambiguities and yields precise charm and bottom quark masses.

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

The paper reanalyzes perturbative QCD corrections to the correlation functions of heavy-quark currents using the Principle of Maximum Conformality combined with a characteristic operator approach. This method absorbs nonconformal terms to produce a scheme- and scale-independent conformal series for the sum-rule moments. The authors then use Padé approximants to estimate the missing fourth-order terms and extract the MS-bar masses from specific moments of the charmed pseudoscalar and bottom vector correlators. These determinations agree with experimental world averages to within less than one standard deviation, providing tighter constraints for particle physics phenomenology.

Core claim

Using the CO-PMC approach on the O(α_s^3) pQCD corrections to M_n,q^{X,th}, the perturbative series becomes strictly conformal and independent of renormalization scheme and scale. Combined with Padé estimates for N4LO, this leads to m_c(m_c) = 1275.8 ± 0.4 MeV from M_{2,c}^P and m_b(m_b) = 4177.0 ± 7.2 MeV from M_{1,b}^V, both consistent with PDG averages within 1σ.

What carries the argument

The characteristic operator (CO) extension of PMC, which simultaneously fixes the effective coupling and quark mass by absorbing β_i and γ_i terms through renormalization group equations to generate a conformal perturbative expansion.

If this is right

  • The extracted quark masses serve as improved inputs for calculations of quarkonium properties and heavy-flavor physics.
  • Scale-independent results allow for more reliable comparisons between theory and experiment in sum-rule analyses.
  • The Padé method provides a systematic way to estimate unknown higher-order perturbative contributions in similar processes.
  • Application to other moments or channels could further refine the mass values or extract other parameters like decay constants.

Where Pith is reading between the lines

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

  • If the CO-PMC procedure generalizes, it could be used to remove perturbative ambiguities in other QCD observables beyond sum rules.
  • The close agreement with PDG suggests that non-perturbative contributions or experimental inputs now limit the precision rather than the perturbative series itself.
  • Future lattice QCD calculations of the same correlators could test the consistency of the perturbative treatment independent of sum-rule assumptions.

Load-bearing premise

The characteristic operator extension of PMC produces a strictly scheme- and scale-independent conformal series, and Padé approximants accurately estimate the unknown N4LO terms without large uncontrolled errors.

What would settle it

A full N4LO calculation of the quarkonium correlation functions that differs substantially from the Padé predictions, or a new experimental determination of the charm or bottom quark mass lying outside the stated uncertainty ranges.

Figures

Figures reproduced from arXiv: 2605.07618 by Hua Zhou, Qing Yu, Xing-Gang Wu.

Figure 1
Figure 1. Figure 1: FIG. 1: The moments [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The moments [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The moments [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The moments [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We reanalyze the perturbative QCD (pQCD) corrections to quarkonium QCD sum rules and extract the heavy quark masses $\overline{m}_{q}(\overline{m}_{q})$ ($q=c,b$). At present, the pQCD corrections to the correlation functions of two heavy-quark pseudoscalar and vector currents at zero momentum transfer, denoted as $M_{n,q}^{X,\rm th}$ ($X = P, V$), are calculated up to the $\mathcal{O}(\alpha_s^3)$ order. These corrections exhibit significant renormalization scheme and scale dependence, which introduces large theoretical uncertainties and deteriorates the precision of heavy quark mass determinations. In this work, we eliminate the renormalization scheme and scale ambiguities in the perturbative part of $M_{n,q}^{X,\rm th}$ by adopting the Principle of Maximum Conformality (PMC) within the characteristic operator (CO) approach. The CO approach, a novel extension of the standard PMC procedure, simultaneously determines the effective coupling $\alpha_s(Q_*)$ and the effective quark mass $\overline{m}_q(Q_*)$. It systematically absorbs the nonconformal $\{\beta_i\}$-terms and $\{\gamma_i\}$-terms via the renormalization group equations, yielding a strictly scheme- and scale-independent conformal perturbative series. Based on the improved PMC conformal series, we further provide reliable estimates for the unknown $\mathrm{N^4LO}$ contributions using the Pad\'e approximation method. The final predicted heavy quark masses in the $\overline{\mathrm{MS}}$ scheme read: $\overline{m}_c(\overline{m}_c)=1275.8\pm 0.4~\text{MeV}$, extracted from the second moment of the charmed pseudoscalar correlator $M_{2,c}^{P}$; and $\overline{m}_b(\overline{m}_b) = 4177.0 \pm 7.2~\text{MeV}$, extracted from the first moment of the bottom vector correlator $M_{1,b}^{V}$. Both results agree well with the PDG world averages with deviations smaller than $1\sigma$.

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

1 major / 1 minor

Summary. The paper reanalyzes pQCD corrections to the two-point correlation functions of heavy-quark pseudoscalar and vector currents up to O(α_s^3) in quarkonium sum rules. It applies the Principle of Maximum Conformality (PMC) extended via the characteristic operator (CO) approach to absorb all β- and γ-dependent nonconformal terms, yielding a strictly scheme- and scale-independent conformal series. Unknown N4LO contributions are estimated via Padé approximants. The charm mass is extracted from the second moment of the pseudoscalar correlator as m_c(m_c)=1275.8±0.4 MeV and the bottom mass from the first moment of the vector correlator as m_b(m_b)=4177.0±7.2 MeV, both in the MS-bar scheme, with results agreeing with PDG averages to better than 1σ.

Significance. If the CO-PMC procedure produces a genuinely conformal series free of residual scale dependence and the Padé estimates prove accurate, the work would deliver high-precision heavy-quark mass determinations with substantially reduced theoretical uncertainties relative to conventional scale-variation analyses. The sub-MeV error on m_c and the close PDG agreement would represent a notable advance for QCD sum-rule phenomenology, provided the error budget is robust.

major comments (1)
  1. [Abstract] Abstract: The final results m_c(m_c)=1275.8±0.4 MeV and m_b(m_b)=4177.0±7.2 MeV are obtained only after replacing the unknown N4LO coefficient in the CO-PMC series by a Padé approximant. No cross-validation is reported (e.g., applying the same Padé procedure to predict the already-known N3LO coefficient and comparing it with its explicit value). Because QCD series are subject to renormalon-induced factorial growth, an untested Padé extrapolation can introduce an error comparable to or larger than the quoted 0.4 MeV uncertainty on m_c; this directly affects the claimed precision and must be addressed.
minor comments (1)
  1. [Abstract] The abstract refers to 'the second moment of the charmed pseudoscalar correlator M_{2,c}^P' and 'the first moment of the bottom vector correlator M_{1,b}^V' without stating the explicit definitions or the rationale for selecting these particular moments over others; a brief clarification would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major point raised below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The final results m_c(m_c)=1275.8±0.4 MeV and m_b(m_b)=4177.0±7.2 MeV are obtained only after replacing the unknown N4LO coefficient in the CO-PMC series by a Padé approximant. No cross-validation is reported (e.g., applying the same Padé procedure to predict the already-known N3LO coefficient and comparing it with its explicit value). Because QCD series are subject to renormalon-induced factorial growth, an untested Padé extrapolation can introduce an error comparable to or larger than the quoted 0.4 MeV uncertainty on m_c; this directly affects the claimed precision and must be addressed.

    Authors: We thank the referee for highlighting this issue. The CO-PMC procedure, through the characteristic operator approach, absorbs all β- and γ-dependent nonconformal terms into the effective coupling and mass scales, producing a strictly conformal perturbative series. Renormalon effects, which drive factorial growth in conventional QCD expansions, are associated with these nonconformal terms and are therefore eliminated; the remaining conformal series exhibits improved convergence properties that make Padé extrapolation more reliable than in the standard case. We acknowledge that an explicit cross-validation—applying the Padé approximant to predict the known N3LO coefficient from lower orders and comparing it to the exact value—was not reported in the original manuscript. We will add this validation (including quantitative comparison) to the revised version, together with a clearer explanation of how the spread among different Padé approximants is used to estimate the N4LO uncertainty and how it is incorporated into the final error budget. These additions will directly address the concern about the robustness of the quoted precision. revision: yes

Circularity Check

0 steps flagged

No circularity in the quark mass extraction via CO-PMC sum rules

full rationale

The paper reorganizes the known O(α_s^3) perturbative corrections to the quarkonium correlator moments using the CO extension of PMC, which by construction absorbs nonconformal β_i and γ_i terms via the RGE to produce a scale- and scheme-independent conformal series. It then applies Padé approximants to estimate the unknown N^4LO coefficient and extracts the MS-bar masses by matching the resulting theoretical moments M_n^th to experimental inputs. This matching step determines the mass values from external data rather than by internal definition or by renaming a fitted parameter as a prediction. No self-definitional reduction, fitted-input-called-prediction, or load-bearing self-citation chain is present; the central claim retains independent content from the data fit and the perturbative reorganization.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or ad-hoc assumptions; the method assumes standard perturbative QCD and the validity of the PMC absorption procedure, but details are unavailable.

pith-pipeline@v0.9.0 · 5689 in / 1243 out tokens · 52475 ms · 2026-05-11T02:30:09.175380+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

114 extracted references · 114 canonical work pages

  1. [1]

    Weinberg, Physica A96, no.1-2, 327-340 (1979)

    S. Weinberg, Physica A96, no.1-2, 327-340 (1979)

    work page 1979

  2. [2]

    Gasser and H

    J. Gasser and H. Leutwyler, Annals Phys.158, 142 (1984)

    work page 1984

  3. [3]

    Pich, Rept

    A. Pich, Rept. Prog. Phys.58, 563-610 (1995)

    work page 1995

  4. [4]

    Weinberg, Trans

    S. Weinberg, Trans. New York Acad. Sci.38, 185-201 (1977)

    work page 1977

  5. [5]

    J. O. Andersen, Q. Yu and H. Zhou, Phys. Rev. D109, 034022 (2024)

    work page 2024

  6. [6]

    Del Debbio and A

    L. Del Debbio and A. Ramos, [arXiv:2101.04762 [hep- lat]]

    work page arXiv

  7. [7]

    Aokiet al.[Flavour Lattice Averaging Group (FLAG)], Eur

    Y. Aokiet al.[Flavour Lattice Averaging Group (FLAG)], Eur. Phys. J. C82, 869 (2022)

    work page 2022

  8. [8]

    D. J. Gross and F. Wilczek, Phys. Rev. Lett.30, 1343- 1346 (1973)

    work page 1973

  9. [9]

    H. D. Politzer, Phys. Rev. Lett.30, 1346-1349 (1973)

    work page 1973

  10. [10]

    V. A. Novikov, L. B. Okun, M. A. Shifman, A. I. Vain- shtein, M. B. Voloshin and V. I. Zakharov, Phys. Rept. 41(1978), 1-133

    work page 1978

  11. [11]

    L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127, 1 (1985)

    work page 1985

  12. [12]

    J. H. K¨ uhn and M. Steinhauser, Nucl. Phys. B619, 588-602 (2001) [erratum: Nucl. Phys. B640, 415-415 (2002)]

    work page 2001

  13. [13]

    J. H. K¨ uhn, M. Steinhauser and C. Sturm, Nucl. Phys. B778, 192-215 (2007)

    work page 2007

  14. [14]

    M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147(1979), 385-447

    work page 1979

  15. [15]

    Maezawa and P

    Y. Maezawa and P. Petreczky, Phys. Rev. D94, 034507 (2016)

    work page 2016

  16. [16]

    Petreczky and J

    P. Petreczky and J. H. Weber, Phys. Rev. D100, 034519 (2019)

    work page 2019

  17. [17]

    Allisonet al.[HPQCD], Phys

    I. Allisonet al.[HPQCD], Phys. Rev. D78, 054513 (2008)

    work page 2008

  18. [18]

    McNeile, C

    C. McNeile, C. T. H. Davies, E. Follana, K. Hornbostel and G. P. Lepage, Phys. Rev. D82, 034512 (2010)

    work page 2010

  19. [19]

    Petreczky and J

    P. Petreczky and J. H. Weber, Eur. Phys. J. C82, 64 (2022)

    work page 2022

  20. [20]

    Nakayama, B

    K. Nakayama, B. Fahy and S. Hashimoto, Phys. Rev. D 94, 054507 (2016)

    work page 2016

  21. [21]

    M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147(1979), 448-518

    work page 1979

  22. [22]

    A. O. G. Kallen and A. Sabry, Kong. Dan. Vid. Sel. Mat. Fys. Med.29(1955)

    work page 1955

  23. [23]

    K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Phys. Lett. B371(1996), 93-98

    work page 1996

  24. [24]

    K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Nucl. Phys. B482(1996), 213-240

    work page 1996

  25. [25]

    Boughezal, M

    R. Boughezal, M. Czakon and T. Schutzmeier, Nucl. Phys. B Proc. Suppl.160(2006), 160-164

    work page 2006

  26. [26]

    Czakon and T

    M. Czakon and T. Schutzmeier, JHEP07(2008), 001

    work page 2008

  27. [27]

    Maier, P

    A. Maier, P. Maierhofer and P. Marquard, Nucl. Phys. B797(2008), 218-242

    work page 2008

  28. [28]

    K. G. Chetyrkin, J. H. Kuhn and C. Sturm, Eur. Phys. J. C48(2006), 107-110

    work page 2006

  29. [29]

    Boughezal, M

    R. Boughezal, M. Czakon and T. Schutzmeier, Phys. Rev. D74(2006), 074006

    work page 2006

  30. [30]

    Maier, P

    A. Maier, P. Maierhofer and P. Marquard, Phys. Lett. B669(2008), 88-91

    work page 2008

  31. [31]

    Maier, P

    A. Maier, P. Maierhofer, P. Marquard and A. V. Smirnov, Nucl. Phys. B824(2010), 1-18

    work page 2010

  32. [32]

    Maier and P

    A. Maier and P. Marquard, Phys. Rev. D97(2018), 056016

    work page 2018

  33. [33]

    A. H. Hoang, V. Mateu and S. Mohammad Zebarjad, Nucl. Phys. B813(2009), 349-369

    work page 2009

  34. [34]

    Y. Kiyo, A. Maier, P. Maierhofer and P. Marquard, Nucl. Phys. B823(2009), 269-287. 20

    work page 2009

  35. [35]

    Greynat and S

    D. Greynat and S. Peris, Phys. Rev. D82(2010), 034030 [erratum: Phys. Rev. D82(2010), 119907]

    work page 2010

  36. [36]

    Boito, V

    D. Boito, V. Mateu and M. V. Rodrigues, JHEP08 (2021), 027

    work page 2021

  37. [37]

    Signer, Phys

    A. Signer, Phys. Lett. B672, 333-338 (2009)

    work page 2009

  38. [38]

    Signer, Phys

    A. Signer, Phys. Lett. B654, 206-214 (2007)

    work page 2007

  39. [39]

    D. J. Gross and A. Neveu, Phys. Rev. D10, 3235 (1974)

    work page 1974

  40. [40]

    B. E. Lautrup, Phys. Lett. B69, 109-111 (1977)

    work page 1977

  41. [41]

    Beneke, Phys

    M. Beneke, Phys. Rept.317, 1-142 (1999)

    work page 1999

  42. [42]

    Dehnadi, A

    B. Dehnadi, A. H. Hoang and V. Mateu, JHEP08, 155 (2015)

    work page 2015

  43. [43]

    Dehnadi, A

    B. Dehnadi, A. H. Hoang, V. Mateu and S. M. Zebarjad, JHEP09, 103 (2013)

    work page 2013

  44. [44]

    Petermann, Helv

    A. Petermann, Helv. Phys. Acta26, 499 (1953)

    work page 1953

  45. [45]

    C. G. Callan, Jr., Phys. Rev. D2, 1541 (1970)

    work page 1970

  46. [46]

    Symanzik, Commun

    K. Symanzik, Commun. Math. Phys.18, 227 (1970)

    work page 1970

  47. [47]

    Peterman, Phys

    A. Peterman, Phys. Rept.53, 157 (1979)

    work page 1979

  48. [48]

    S. J. Brodsky and X. G. Wu, Phys. Rev. D85, 034038 (2012)

    work page 2012

  49. [49]

    S. J. Brodsky and X. G. Wu, Phys. Rev. Lett.109, 042002 (2012)

    work page 2012

  50. [50]

    Mojaza, S

    M. Mojaza, S. J. Brodsky and X. G. Wu, Phys. Rev. Lett.110, 192001 (2013)

    work page 2013

  51. [51]

    S. J. Brodsky, M. Mojaza and X. G. Wu, Phys. Rev. D 89, 014027 (2014)

    work page 2014

  52. [52]

    X. G. Wu, Y. Ma, S. Q. Wang, H. B. Fu, H. H. Ma, S. J. Brodsky and M. Mojaza, Rep. Prog. Phys.78, 126201 (2015)

    work page 2015

  53. [53]

    X. G. Wu, S. Q. Wang and S. J. Brodsky, Front. Phys. 11, 111201 (2016)

    work page 2016

  54. [54]

    X. G. Wu, J. M. Shen, B. L. Du, X. D. Huang, S. Q. Wang and S. J. Brodsky, Prog. Part. Nucl. Phys. 108, 103706 (2019)

    work page 2019

  55. [55]

    J. Yan, S. J. Brodsky, L. Di Giustino, P. G. Ratcliffe, S. Wang, S. Q. Wang, X. Wu and X. G. Wu, Symmetry 17, 411 (2025)

    work page 2025

  56. [56]

    S. J. Brodsky and L. Di Giustino, Phys. Rev. D86, 085026 (2012)

    work page 2012

  57. [57]

    X. C. Zheng, X. G. Wu, S. Q. Wang, J. M. Shen and Q. L. Zhang, JHEP10, 117 (2013)

    work page 2013

  58. [58]

    J. M. Shen, X. G. Wu, B. L. Du and S. J. Brodsky, Phys. Rev. D95, 094006 (2017)

    work page 2017

  59. [59]

    X. G. Wu, J. M. Shen, B. L. Du and S. J. Brodsky, Phys. Rev. D97, 094030 (2018)

    work page 2018

  60. [60]

    J. M. Shen, B. H. Qin, J. Yan, S. Q. Wang and X. G. Wu, JHEP07, 109 (2023)

    work page 2023

  61. [61]

    S. Q. Wang, C. Q. Luo, X. G. Wu, J. M. Shen and L. Di Giustino, JHEP09, 137 (2022)

    work page 2022

  62. [62]

    Q. Yu, H. Zhou, X. D. Huang, J. M. Shen and X. G. Wu, Chin. Phys. Lett.39, 071201 (2022)

    work page 2022

  63. [63]

    H. Y. Bi, X. G. Wu, Y. Ma, H. H. Ma, S. J. Brodsky and M. Mojaza, Phys. Lett. B748, 13-18 (2015)

    work page 2015

  64. [64]

    S. Q. Wang, X. G. Wu, X. C. Zheng, G. Chen and J. M. Shen, J. Phys. G41, 075010 (2014)

    work page 2014

  65. [65]

    Q. Yu, X. G. Wu, S. Q. Wang, X. D. Huang, J. M. Shen and J. Zeng, Chin. Phys. C43, 093102 (2019)

    work page 2019

  66. [66]

    X. D. Huang, X. G. Wu, J. Zeng, Q. Yu, X. C. Zheng and S. Xu, Phys. Rev. D101, 114024 (2020)

    work page 2020

  67. [67]

    X. D. Huang, X. G. Wu, X. C. Zheng, J. Yan, Z. F. Wu and H. H. Ma, Chin. Phys. C48, 053113 (2024)

    work page 2024

  68. [68]

    S. Y. Ma, X. D. Huang, X. C. Zheng and X. G. Wu, Chin. Phys. Lett.41, 101201 (2024)

    work page 2024

  69. [69]

    J. Yan, X. G. Wu, J. M. Shen, X. D. Huang and Z. F. Wu, JHEP04, 184 (2025)

    work page 2025

  70. [70]

    B. L. Du, X. G. Wu, J. M. Shen and S. J. Brodsky, Eur. Phys. J. C79, 182 (2019)

    work page 2019

  71. [71]

    J. M. Shen, Z. J. Zhou, S. Q. Wang, J. Yan, Z. F. Wu, X. G. Wu and S. J. Brodsky, Eur. Phys. J. C83, no.4, 326 (2023)

    work page 2023

  72. [72]

    K. G. Chetyrkin, Phys. Lett. B404, 161-165 (1997)

    work page 1997

  73. [73]

    J. A. M. Vermaseren, S. A. Larin and T. van Ritbergen, Phys. Lett. B405, 327-333 (1997)

    work page 1997

  74. [74]

    K. G. Chetyrkin, Nucl. Phys. B710, 499 (2005)

    work page 2005

  75. [75]

    Czakon, Nucl

    M. Czakon, Nucl. Phys. B710, 485 (2005)

    work page 2005

  76. [76]

    P. A. Baikov, K. G. Chetyrkin and J. H. K¨ uhn, JHEP 10, 076 (2014)

    work page 2014

  77. [77]

    P. A. Baikov, K. G. Chetyrkin and J. H. K¨ uhn, Phys. Rev. Lett.118, no.8, 082002 (2017)

    work page 2017

  78. [78]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, JHEP02, 090 (2017)

    work page 2017

  79. [79]

    B. L. Ioffe, Prog. Part. Nucl. Phys.56(2006), 232-277

    work page 2006

  80. [80]

    Narison and R

    S. Narison and R. Tarrach, Phys. Lett. B125(1983), 217-222

    work page 1983

Showing first 80 references.