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REVIEW 3 major objections 5 minor 48 references

Leading-renormalon-free Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks up to Five Loops in QCD

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The leading infrared-renormalon divergence of the heavy-quark pole mass lies entirely in the QCD trace anomaly, so the subtracted σ-mass is leading-renormalon-free through five loops.

desk verdict A useful five-loop mass relation that is probably right, but the central renormalon-freedom claim rests on an identity from the companion paper that this text does not prove. read the letter →

arxiv 2509.10786 v1 pith:M5ETL5NB submitted 2025-09-13 hep-ph hep-lathep-th

Long Chen , Cong Zhao This is my paper · ORCID
classification hep-phhep-lathep-th
keywords infraredrenormalonspolemasstraceanomalyheavyquarkmassessigma-massfive-loopQCDbetafunctionanomalousdimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the leading infrared-renormalon ambiguity of the perturbative heavy-quark pole mass is carried entirely by the trace-anomaly contribution, the piece $\mu\, dm_{\mathrm{os}}/d\mu$ that appears when the on-shell mass is viewed as a function of the dimensional-regularization scale. It therefore defines a trace-anomaly-subtracted $\sigma$-mass, $m_\sigma = m_{\mathrm{os}} - \mu\, dm_{\mathrm{os}}/d\mu$, and claims this mass is simultaneously scheme- and scale-invariant and free of the leading renormalon. The central result is eq. (17), a conversion formula written only in terms of the QCD $\beta$-function, the quark-mass anomalous dimension $\gamma_m$, and the pole-to-$\overline{\mathrm{MS}}$ conversion factor $C_m$, together with its five-loop numerical evaluation, eq. (20). If correct, precision heavy-quark physics gains a process-independent short-distance mass that avoids the leading renormalon problem of the pole mass.

What carries the argument

The load-bearing object is the trace-anomaly contribution $\mu\, dm_{\mathrm{os}}/d\mu$ to the pole mass, identified through the dimensionally regulated operator identity (2) that equates the bare amputated matrix element of the gluon trace-anomaly operator with the $\hat\mu$-derivative of the bare quark self-energy. The second load-bearing ingredient is the linear-$\mu$ property of the leading renormalon in the pole-to-$\overline{\mathrm{MS}}$ conversion factor $C_m$, which ensures that the leading renormalon of the trace-anomaly piece equals the leading renormalon of $m_{\mathrm{os}}$ itself. The final mechanism is formula (17), which turns this identification into an explicit conversion between $m_\sigma$ and the on-shell and $\overline{\mathrm{MS}}$ masses using only $\beta$, $\gamma_m$, and the logarithmic coupling-derivative of $C_m$; because $\beta$ starts at $O(\alpha_s)$, the conversion at $O(\alpha_s^N)$ needs $C_m$ only to $O(\alpha_s^{N-1})$.

What would settle it

Compute the Borel transform of the $m_\sigma/m_{\mathrm{os}}$ series (20) and locate its closest singularity: the claim predicts no pole at $u=1/2$ after the trace-anomaly subtraction, so a residual leading singularity there would refute it. Alternatively, an independent evaluation of the 'classical' piece $m_B\,\partial m_{\mathrm{os}}/\partial m_B$ at high loop order that found a nonvanishing leading renormalon residue would contradict eq. (10).

Watch

Extended reading notes

Core claim

The discovery is that the leading infrared-renormalon terms in the pole mass $m_{\mathrm{os}}$ coincide with the leading infrared-renormalon terms in the trace-anomaly contribution $\mu\, \partial m_{\mathrm{os}}/\partial\mu$, so subtracting that contribution removes the leading renormalon without touching the classical fermion-mass part $m_B \partial m_{\mathrm{os}}/\partial m_B$. The proof starts from the operator identity (2) that ties the bare amputated trace-anomaly matrix element to the $\hat\mu$-derivative of the bare quark self-energy, passes through the on-shell renormalization conditions, and uses the established linear-$\mu$ behavior of the leading renormalon in the pole-to-$\overline{\mathrm{MS}}$ conversion factor. The resulting $\sigma$-mass is then expressed through $Z_\sigma = m_\sigma/m_{\mathrm{os}} = (1+2\beta\, \partial \ln C_m/\partial\ln\alpha_s)/(1-2\gamma_m)$, evaluated to five loops in eq. (20) and to four loops as $m_\sigma/m$ in eq. (21).

Load-bearing premise

The paper assumes without proof here the operator identity (2) equating the bare amputated trace-anomaly matrix element to $\hat\mu\,\partial\Sigma_B/\partial\hat\mu$ prior to the on-shell limit; if that identity misses scheme- or gauge-dependent terms, or needs extra counterterms on shell, the identification of the leading renormalon with the trace-anomaly piece fails.

Editorial extensions

If this is right

  • The $\sigma$-mass is expected to be free of the leading $u=1/2$ renormalon while preserving the scheme- and scale-invariance of the pole mass, making it usable as a process-independent heavy-quark mass.
  • The five-loop series in eq. (20) supplies the $m_\sigma/m_{\mathrm{os}}$ conversion at the on-shell scale, with errors set by the per-mille-level numerical constants of the four-loop pole-to-$\overline{\mathrm{MS}}$ relation.
  • The four-loop $m_\sigma/m$ relation in eq. (21) has smaller perturbative coefficients than the usual pole-to-$\overline{\mathrm{MS}}$ series, since the leading renormalon is absent.
  • Updated central values follow for the top and bottom $\sigma$-masses: $m_\sigma^t = 158.67 \pm 0.29$ GeV and $m_\sigma^b = 3.97^{+0.08}_{-0.07}$ GeV from the input pole and $\overline{\mathrm{MS}}$ masses.
  • Relations from $m_\sigma$ to other short-distance mass schemes (kinetic, PS, MSR, renormalon-subtracted) can be derived to three or four loops wherever those masses are already related to the on-shell or $\overline{\mathrm{MS}}$ definitions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct numerical check of the claim would be to compute the Borel transform of the five-loop series in eq. (20) and test whether the leading $u=1/2$ singularity vanishes; the finite coefficients shown in the paper do not by themselves prove renormalon freedom.
  • The same trace-anomaly subtraction logic could, in principle, define renormalon-free mass-like parameters in other gauge theories where the operator identity (2) has an analogue, although the paper does not make that extension.
  • The removal of the leading renormalon suggests that scale-setting procedures applied to $m_\sigma$-based observables should show less residual-scale sensitivity than their pole-mass counterparts, though the paper does not quantify that comparison.
  • Once a five-loop pole-to-$\overline{\mathrm{MS}}$ conversion becomes available, the same formula (17) would immediately produce a six-loop $m_\sigma/m_{\mathrm{os}}$ series using the known five-loop $\beta$ and $\gamma_m$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims that the leading infrared renormalon in the perturbative pole mass of a heavy quark resides entirely in the contribution from the trace anomaly of the energy-momentum tensor. On this basis, the recently proposed trace-anomaly-subtracted sigma-mass is asserted to be free of the leading renormalon. The authors derive a formula (Eq. (17)) connecting m_sigma/m_os to the QCD beta-function, the mass anomalous dimension gamma_m, and the pole-to-MS conversion factor C_m, and use known four- and five-loop results to present five-loop numerical relations for m_sigma/m_os and four-loop relations for m_sigma/m. They also update the sigma-masses of the top and bottom quarks.

Significance. If the central claim holds, the sigma-mass is a promising process-independent heavy-quark mass definition with better perturbative convergence than the pole mass. The RG derivation leading to Eqs. (14) and (17) is internally consistent and elegant, and the five-loop numerical results with documented uncertainties are a useful contribution to the literature. The paper also makes explicit falsifiable predictions for the on-shell matrix element of the MS-renormalized gluon field-strength squared (Eq. (19)) and provides compact conversion relations that can be used in high-energy applications. However, the renormalon-freedom conclusion is conditional on an operator identity taken from a companion paper, and one step in the derivation is written in a way that is misleading if read literally.

major comments (3)
  1. [Section 1, Eqs. (1)-(2)] The central renormalon-freedom claim is conditional on the operator identity (1)/(2) taken without proof from the companion paper [18]. In particular, the passage from the off-shell identity (2) in a generic covariant gauge to the on-shell Landau-gauge identity (1) is asserted but not derived; the gauge-fixing operator in Eq. (2) could in principle contribute in the on-shell limit. Moreover, the normalization of the identity, including any implicit operator renormalization factor, must be consistent with the Z_psi factor appearing in Eq. (4), and no such consistency check is provided here. Because Eq. (10) and the numerical results in Section 3 inherit this identity, the paper's main conclusion is not self-contained and is only as solid as [18].
  2. [Section 1, Eq. (9)] The step leading to Eq. (9) is not rigorously justified. The statement '∂Z_m/∂μ = 0 = ∂m/∂μ' is false if μ is the MS renormalization scale, since Z_m depends on μ through α_s(μ) and the MS mass m(μ) runs with μ. The intended reading appears to be a partial derivative at fixed renormalized parameters, but this is not what the notation conveys. This step is load-bearing because it identifies the trace-anomaly contribution with m μ∂C_m/∂μ; a correct derivation must use the chain rule together with the RG equations and must specify which variables are held fixed. Without this clarification, the identification is not established.
  3. [Section 1, Eq. (10)] The 'linear-μ' property of the leading IR renormalon is used to assert μ∂C_m/∂μ|_{LIR} = C_m|_{LIR}, which is the key cancellation that makes the subtraction work. This property is cited to Refs. [14-16,20] but is not stated precisely. The equality requires that the leading renormalon contribution to C_m is exactly proportional to μ/m with no other μ-dependence; a precise statement and a brief derivation (or a specific quotation from the literature) would make the argument robust and easier to verify.
minor comments (5)
  1. [Section 1, Eq. (1)] The notation 'ampu.' is used without definition; it should be defined as 'amputated' on first use.
  2. [Section 1, paragraph after Eq. (3)] The text says 'If we boldly assume that one can exchange the operation ordering' and then Eq. (6) is presented as a rigorous derivation. The wording is inconsistent; Eq. (6) should be framed as a proof, not as relying on a bold assumption.
  3. [Section 3, Eqs. (20)-(21)] The numerical results are evaluated at a specific renormalization scale (μ=m_os for Eq. (20) and μ=m for Eq. (21)) but this is stated only in prose; it should be repeated in the equation captions or the text directly preceding them. It should also be stated whether the quoted errors in the five-loop coefficients include only the uncertainties of the four-loop C_m or also those of beta and gamma_m.
  4. [Section 3, text after Eq. (20)] The phrase 'one loop-order less' should read 'one loop order less'.
  5. [Footnote 3] The reference to Eq. (2.8) of Ref. [9] is cryptic and does not clearly explain why that formula cannot be applied here; the sentence should be expanded or removed.

Circularity Check

1 steps flagged · score 4.0 of 10

Renormalon-freedom proof imports its key operator identity from same-author companion paper [18] without proof here; five-loop conversion itself is externally anchored and not fitted.

  1. self citation load bearing [Section 1, eqs. (1)-(2) and the text before them; used in eqs. (6), (10), (14), (17)-(21)]
    "Regarding the trace-anomaly contribution to the perturbative pole mass in QCD ... one of us has derived the following relation [18] to any loop orders, ... (1) ... We shall take eq. (1) and/or eq. (2) as the starting point of the following discussion on the leading IR-renormalon terms in the quantum trace-anomaly contribution ... so-defined at the on-shell limit."

    The paper's central result, that the trace-anomaly contribution carries exactly the leading IR renormalon of m_os, is obtained by combining eq. (1)/(2) with the external linear-mu property. Eq. (1)/(2) is not proved in this text; it is imported from companion paper [18] by the same first author. The 'rigorous derivation' in eqs. (4)-(6) starts from eq. (4), which is already the Z_psi-renormalized version of the same imported identity, so it does not supply an independent check. If the normalization of the identity in [18] were off by a Z_psi-type factor, or if a gauge/equation-of-motion remainder survived the on-shell Landau-gauge limit, eqs. (6), (10), (14), (17) and the five-loop numbers (20)-(21) would be rescaled or shifted, and the renormalon-freedom of m_sigma would not follow.

full rationale

The derivation chain is: (i) companion-paper identity eq. (1)/(2) equates the on-shell trace-anomaly matrix element with mu dSigma_B/dmu; (ii) eqs. (3)-(6) turn this into mu dm_os/dmu; (iii) the external linear-mu property of the leading renormalon in C_m [14-16,20] gives mu dm_os/dmu|_{LIR} = m_os|_{LIR}, proving renormalon-freedom of m_sigma = m_os - mu dm_os/dmu. Step (iii) is a legitimate use of an external, mature result. Step (ii) is algebra once (i) is granted. The weak point is step (i): it is not proved in this text, and the 'rigorous derivation' of eq. (3) starts from eq. (4), which is just the Z_psi-renormalized form of the same imported identity. Because the same first author is the source and no machine-checked or externally reproduced verification is supplied, this is a load-bearing self-citation rather than an independent mathematical fact. The five-loop coefficients in eqs. (20)-(21), however, are not fitted: they follow from formula (17)/(18) using literature values of beta, gamma_m, and the four-loop C_m, so no fitted-input-called-prediction circularity occurs. The sigma-mass is an operator-defined mass, not a renaming of a known result, though its subtraction idea overlaps with renormalon-subtracted masses. Overall, the headline renormalon-freedom claim is conditional on the companion-paper identity, but the five-loop computation itself is externally anchored, giving partial circularity rather than a fully forced result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No numerical parameter is fitted in this paper; the derived five-loop coefficients use published beta, gamma_m, and four-loop pole-to-MS conversion factors as inputs. The main assumed ingredients are the self-cited operator identity (1) and the standard linear-mu property of the leading renormalon. The sigma-mass is a newly introduced definition with no independent falsifiable handle outside the framework.

assumptions (5)
  • domain assumption Bare amputated matrix element of the EMT trace-anomaly operator equals mu d Sigma_B / d mu (eqs. (1)-(2), from ref [18])
    Self-cited companion result, not re-derived here; load-bearing for the identification of the trace-anomaly contribution with the scale derivative of the pole mass.
  • domain assumption Leading IR-renormalon terms in C_m depend on mu only linearly (refs [14-16,20])
    Standard renormalon property, cited; essential for concluding that mu d C_m / d mu selects the full leading renormalon.
  • standard math Euler homogeneity m_os = mu d m_os / d mu + m_B d m_os / d m_B (eq. (15))
    Mass-dimensional analysis; requires m_os to be a homogeneous function of degree one in mu and m_B.
  • domain assumption On-shell renormalization conditions Sigma_R = 0 and d Sigma_R / d slash-p = 0 at slash-p = m_os (eq. (5))
    Standard on-shell renormalization conditions used to justify eq. (6).
  • ad hoc to paper C_m is expressed as a function of alpha_s and mu/m, not mu/m_os, with logarithms rewritten via iteration of m_os = m C_m (eq. (8))
    This rewriting is what makes the partial derivative mu d C_m / d mu non-trivial; the RG equation (11) and the final formula (17) depend on this choice.
invented entities (1)
  • sigma-mass m_sigma (trace-anomaly-subtracted heavy-quark mass)
    purpose: A heavy-quark mass definition that is scheme/scale-invariant and free of the leading IR renormalon; used as an alternative to pole/MS masses in high-precision applications.
    Introduced in companion paper [18]; not a new force or particle but a new composite mass definition. It has no independent falsifiable prediction beyond shifting extracted quark masses; its numerical values for t and b quarks are derived from existing inputs.

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Cite this review

Pith. "Pith review of Leading-renormalon-free Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks up to Five Loops in QCD." pith.science (2026). https://pith.science/paper/M5ETL5NB

@misc{pith2026250910786,
  author       = {Pith},
  title        = {Pith review of: Leading-renormalon-free Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks up to Five Loops in QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5ETL5NB}},
  note         = {Machine review of arXiv:2509.10786}
}
abstract

We demonstrate that the leading IR-renormalon divergence in the perturbative pole mass of a massive quark resides entirely in the contribution from the trace anomaly of the energy-momentum tensor in QCD. Consequently, the recently proposed trace-anomaly-subtracted $\sigma$-mass definition for heavy quarks is not only scheme- and scale-invariant, but also free from the leading IR-renormalon ambiguity. We further derive a formula connecting this $\sigma$-mass to the perturbative pole mass, solely in terms of the QCD $\beta$-function, quark-mass anomalous dimension $\gamma_m$ and a proper rewritten form of the pole-to-$\overline{\mathrm{MS}}$ mass conversion factor. Utilizing this formula along with the ingredients available in the literature, we present the explicit five-loop result for the perturbative relationship between the $\sigma$-mass and the perturbative pole mass in QCD under the approximation of keeping only a single quark massive. Given the theoretical merits of this mass definition and the availability of high-precision conversion relations, we encourage its application to high-energy processes with heavy quarks, e.g., $H \rightarrow b\bar{b} + X_{\mathrm{QCD}}$, and to current-current correlators used in determining heavy-quark masses and decay widths.

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Works this paper leans on

48 extracted references · 14 canonical work pages

  1. [18]

    L. Chen, Z. Li, and M. Niggetiedt,On-shell Matrix Elements of the EMT Trace in Gauge Theories and Heavy Quark Masses,arXiv:2509.03580 [hep-ph]

    work page arXiv

  2. [1]

    ’t Hooft,Can We Make Sense Out of Quantum Chromodynamics?,Subnucl

    G. ’t Hooft,Can We Make Sense Out of Quantum Chromodynamics?,Subnucl. Ser.15(1979) 943

    work page 1979

  3. [2]

    Parisi,Singularities of the Borel Transform in Renormalizable Theories,Phys

    G. Parisi,Singularities of the Borel Transform in Renormalizable Theories,Phys. Lett. B76(1978) 65–66

    work page 1978

  4. [3]

    Parisi,On Infrared Divergences,Nucl

    G. Parisi,On Infrared Divergences,Nucl. Phys. B150(1979) 163–172

    work page 1979

  5. [4]

    David,On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion,Nucl

    F. David,On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion,Nucl. Phys. B234(1984) 237–251

    work page 1984

  6. [5]

    A. H. Mueller,On the Structure of Infrared Renormalons in Physical Processes at High-Energies,Nucl. Phys. B250(1985) 327–350

    work page 1985

  7. [6]

    R. J. Crewther,Nonperturbative evaluation of the anomalies in low-energy theorems,Phys. Rev. Lett. 28(1972) 1421

    work page 1972

  8. [7]

    M. S. Chanowitz and J. R. Ellis,Canonical Anomalies and Broken Scale Invariance,Phys. Lett. B40 (1972) 397–400

    work page 1972

Show all 48 references
  1. [8]

    M. S. Chanowitz and J. R. Ellis,Canonical Trace Anomalies,Phys. Rev. D7(1973) 2490–2506

    1973

  2. [9]

    S. L. Adler, J. C. Collins, and A. Duncan,Energy-Momentum-Tensor Trace Anomaly in Spin 1/2 Quantum Electrodynamics,Phys. Rev. D15(1977) 1712

    1977

  3. [10]

    J. C. Collins, A. Duncan, and S. D. Joglekar,Trace and Dilatation Anomalies in Gauge Theories, Phys. Rev. D16(1977) 438–449

    1977

  4. [11]

    N. K. Nielsen,The Energy Momentum Tensor in a Nonabelian Quark Gluon Theory,Nucl. Phys. B 120(1977) 212–220

    1977

  5. [12]

    J. C. Breckenridge, M. J. Lavelle, and T. G. Steele,The Nielsen identities for the two point functions of QED and QCD,Z. Phys. C65(1995) 155–164,arXiv:hep-th/9407028

    1995 arXiv

  6. [13]

    A. S. Kronfeld,The Perturbative pole mass in QCD,Phys. Rev. D58(1998) 051501, arXiv:hep-ph/9805215

    1998 arXiv

  7. [14]

    I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev, and A. I. Vainshtein,The Pole mass of the heavy quark. Perturbation theory and beyond,Phys. Rev. D50(1994) 2234–2246,arXiv:hep-ph/9402360

    1994 arXiv

  8. [15]

    Beneke and V

    M. Beneke and V. M. Braun,Heavy quark effective theory beyond perturbation theory: Renormalons, the pole mass and the residual mass term,Nucl. Phys. B426(1994) 301–343,arXiv:hep-ph/9402364

    1994 arXiv

  9. [16]

    Beneke,More on ambiguities in the pole mass,Phys

    M. Beneke,More on ambiguities in the pole mass,Phys. Lett. B344(1995) 341–347, arXiv:hep-ph/9408380

    1995 arXiv

  10. [17]

    M. C. Smith and S. S. Willenbrock,Top quark pole mass,Phys. Rev. Lett.79(1997) 3825–3828, arXiv:hep-ph/9612329

    1997 arXiv

  11. [19]

    Tarrach,The Pole Mass in Perturbative QCD,Nucl

    R. Tarrach,The Pole Mass in Perturbative QCD,Nucl. Phys. B183(1981) 384–396

    1981

  12. [20]

    Beneke,Renormalons,Phys

    M. Beneke,Renormalons,Phys. Rept.317(1999) 1–142,arXiv:hep-ph/9807443

    1999 arXiv

  13. [21]

    Melnikov and T

    K. Melnikov and T. v. Ritbergen,The Three loop relation between the MS-bar and the pole quark masses,Phys. Lett. B482(2000) 99–108,arXiv:hep-ph/9912391. 8

    2000 arXiv

  14. [22]

    K. G. Chetyrkin and M. Steinhauser,The Relation between the MS-bar and the on-shell quark mass at order alpha(s)**3,Nucl. Phys. B573(2000) 617–651,arXiv:hep-ph/9911434

    2000 arXiv

  15. [23]

    Marquard, A

    P. Marquard, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser,Quark Mass Relations to Four-Loop Order in Perturbative QCD,Phys. Rev. Lett.114no. 14, (2015) 142002,arXiv:1502.01030 [hep-ph]

    2015 arXiv

  16. [24]

    Marquard, A

    P. Marquard, A. V. Smirnov, V. A. Smirnov, M. Steinhauser, and D. Wellmann, MS-on-shell quark mass relation up to four loops in QCD and a general SU(N)gauge group,Phys. Rev. D94no. 7, (2016) 074025,arXiv:1606.06754 [hep-ph]

    2016 arXiv

  17. [25]

    A. L. Kataev and V. S. Molokoedov,Multiloop contributions to the on-shell- MSheavy quark mass relation in QCD and the asymptotic structure of the corresponding series: the updated consideration, Eur. Phys. J. C80no. 12, (2020) 1160,arXiv:1807.05406 [hep-ph]

    2020 arXiv

  18. [26]

    Herren and M

    F. Herren and M. Steinhauser,Version 3 of RunDec and CRunDec,Comput. Phys. Commun.224 (2018) 333–345,arXiv:1703.03751 [hep-ph]

    2018 arXiv

  19. [27]

    Baikov, K

    P. Baikov, K. Chetyrkin, and J. K¨ uhn,Five-Loop Running of the QCD coupling constant,Phys. Rev. Lett.118no. 8, (2017) 082002,arXiv:1606.08659 [hep-ph]

    2017 arXiv

  20. [28]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt,The five-loop beta function of Yang-Mills theory with fermions,JHEP02(2017) 090,arXiv:1701.01404 [hep-ph]

    2017 arXiv

  21. [29]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard, and Y. Schroder,The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge,JHEP10(2017) 166,arXiv:1709.07718 [hep-ph]

    2017 arXiv

  22. [30]

    P. A. Baikov, K. G. Chetyrkin, and J. H. K¨ uhn,Quark Mass and Field Anomalous Dimensions to O(α5 s),JHEP10(2014) 076,arXiv:1402.6611 [hep-ph]

    2014 arXiv

  23. [31]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard, and Y. Schr¨ oder,Five-loop quark mass and field anomalous dimensions for a general gauge group,JHEP01(2017) 081,arXiv:1612.05512 [hep-ph]

    2017 arXiv

  24. [32]

    P. A. Baikov, K. G. Chetyrkin, and J. H. K¨ uhn,Five-loop fermion anomalous dimension for a general gauge group from four-loop massless propagators,JHEP04(2017) 119,arXiv:1702.01458 [hep-ph]

    2017 arXiv

  25. [33]

    I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev, and A. I. Vainshtein,Sum rules for heavy flavor transitions in the SV limit,Phys. Rev. D52(1995) 196–235,arXiv:hep-ph/9405410

    1995 arXiv

  26. [34]

    I. I. Y. Bigi, M. A. Shifman, N. Uraltsev, and A. I. Vainshtein,High power n of m(b) in beauty widths and n=5 —>infinity limit,Phys. Rev. D56(1997) 4017–4030,arXiv:hep-ph/9704245

    1997 arXiv

  27. [35]

    Czarnecki, K

    A. Czarnecki, K. Melnikov, and N. Uraltsev,NonAbelian dipole radiation and the heavy quark expansion,Phys. Rev. Lett.80(1998) 3189–3192,arXiv:hep-ph/9708372

    1998 arXiv

  28. [36]

    M. Fael, K. Sch¨ onwald, and M. Steinhauser,Kinetic Heavy Quark Mass to Three Loops,Phys. Rev. Lett.125no. 5, (2020) 052003,arXiv:2005.06487 [hep-ph]

    2020 arXiv

  29. [37]

    Beneke,A Quark mass definition adequate for threshold problems,Phys

    M. Beneke,A Quark mass definition adequate for threshold problems,Phys. Lett. B434(1998) 115–125,arXiv:hep-ph/9804241

    1998 arXiv

  30. [38]

    A. H. Hoang, A. Jain, I. Scimemi, and I. W. Stewart,Infrared Renormalization Group Flow for Heavy Quark Masses,Phys. Rev. Lett.101(2008) 151602,arXiv:0803.4214 [hep-ph]

    2008 arXiv

  31. [39]

    A. H. Hoang, Z. Ligeti, and A. V. Manohar,B decay and the Upsilon mass,Phys. Rev. Lett.82(1999) 277–280,arXiv:hep-ph/9809423. 9

    1999 arXiv

  32. [40]

    A. H. Hoang, A. Jain, C. Lepenik, V. Mateu, M. Preisser, I. Scimemi, and I. W. Stewart,The MSR mass and theO(Λ QCD)renormalon sum rule,JHEP04(2018) 003,arXiv:1704.01580 [hep-ph]

    2018 arXiv

  33. [41]

    Pineda,Determination of the bottom quark mass from the Upsilon(1S) system,JHEP06(2001) 022,arXiv:hep-ph/0105008

    A. Pineda,Determination of the bottom quark mass from the Upsilon(1S) system,JHEP06(2001) 022,arXiv:hep-ph/0105008

    2001 arXiv

  34. [42]

    Komijani,A discussion on leading renormalon in the pole mass,JHEP08(2017) 062, arXiv:1701.00347 [hep-ph]

    J. Komijani,A discussion on leading renormalon in the pole mass,JHEP08(2017) 062, arXiv:1701.00347 [hep-ph]

    2017 arXiv

  35. [43]

    Martinelli, C

    G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas,A General method for nonperturbative renormalization of lattice operators,Nucl. Phys. B445(1995) 81–108, arXiv:hep-lat/9411010

    1995 arXiv

  36. [44]

    Aokiet al.,Non-perturbative renormalization of quark bilinear operators and B(K) using domain wall fermions,Phys

    Y. Aokiet al.,Non-perturbative renormalization of quark bilinear operators and B(K) using domain wall fermions,Phys. Rev. D78(2008) 054510,arXiv:0712.1061 [hep-lat]

    2008 arXiv

  37. [45]

    Sturm, Y

    C. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda, and A. Soni,Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D80(2009) 014501,arXiv:0901.2599 [hep-ph]

    2009 arXiv

  38. [46]

    Boyle, L

    P. Boyle, L. Del Debbio, and A. Khamseh,Massive momentum-subtraction scheme,Phys. Rev. D95 no. 5, (2017) 054505,arXiv:1611.06908 [hep-lat]

    2017 arXiv

  39. [47]

    Beneke,Pole mass renormalon and its ramifications,Eur

    M. Beneke,Pole mass renormalon and its ramifications,Eur. Phys. J. ST230no. 12-13, (2021) 2565–2579,arXiv:2108.04861 [hep-ph]

    2021 arXiv

  40. [48]

    K. G. Chetyrkin, J. H. Kuhn, and M. Steinhauser,RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses,Comput. Phys. Commun.133(2000) 43–65, arXiv:hep-ph/0004189. [49]Particle Data GroupCollaboration, S. Navaset al.,Review of particle ...

    2000 arXiv

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Reviewed August 15, 2026 · model on record in the stance chip above.