Factorial growth in perturbation theory, power corrections: precise extraction of quark masses and $\alpha_\text{s}$

Andreas S. Kronfeld

REVIEW 3 major objections 8 minor 1 cited by

Factorial growth in perturbation theory, power corrections: precise extraction of quark masses and $\alpha_\text{s}$

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that the renormalization-group constraint fixes the factorially growing high-order coefficients of a QCD series from the first few terms, so the divergent tail can be subtracted and Borel-summed, leaving truncation errors…

desk verdict A clean, honest proceedings summary of the author's own MRS program; the exact inversion is nice, but the all-orders extrapolation from four coefficients is still an unproven consistency assumption, not a demonstrated result. read the letter →

arxiv 2505.20531 v1 pith:X7UQXHSO submitted 2025-05-26 hep-ph hep-th

Andreas S. Kronfeld This is my paper

classification hep-phhep-th

keywords factorialgrowthrenormalonsminimalrenormalonsubtractionpowercorrectionsquarkmassesstrongcouplingconstantstaticenergyBjorkensumrule

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

Quantum chromodynamics predicts observables through asymptotic series whose coefficients grow factorially, so truncating the series at any finite order leaves an error that cannot be estimated from the terms one has. This paper derives a way to close that gap: the renormalization-group constraint, namely that a physical quantity cannot depend on the artificial scale $\mu$ introduced by modified minimal subtraction, fixes the factorial growth already at low orders. Given the first four computed coefficients, the factorially growing tail can be reconstructed, separated off, and Borel-summed, leaving a short subtracted series whose truncation error is controlled. The same construction turns the renormalon ambiguity into the power correction $(\Lambda/Q)^p$ that has to be fitted anyway. If the derivation is right, determinations of $\alpha_s$ and of quark masses from the static energy, the pole mass, and the polarized Bjorken sum rule gain a calculable uncertainty instead of an unknown asymptotic tail.

What carries the argument

The central object is the exact inversion of the renormalization-group constraint: the lower-triangular matrix equation $\boldsymbol{f}^{(p)}=Q^{(p)}\boldsymbol{r}$ of eq. (9) is solved row by row to give eq. (10), where the factor $(2\beta_0/p)^l\Gamma(l+1+pb)/\Gamma(1+pb)$ carries the factorial growth and the strength is a mildly $l$-dependent weighted sum of low-order coefficients. Truncating that sum at the last known order defines what the paper calls minimal renormalon subtraction: eq. (14) splits the series into a subtracted part and a tail whose Borel sum is evaluated in the principal-value sense, with the residual ambiguity forming a power term that merges into the fitted nonperturbative constant. The algebra is performed in a coupling scheme chosen so the $\beta$ function takes a simple form, called the geometric scheme, and the results are converted back to standard schemes for phenomenology.

What would settle it

Compute the five-loop coefficient for the static energy, the pole mass, or the polarized Bjorken sum rule and compare it with the range predicted by eq. (11) using only the first four coefficients; a disagreement larger than the scheme variation, or a residual scale dependence that does not shrink after summation, would show that the low-order terms do not fix the tail.

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Extended reading notes

Core claim

The paper's claim is that the relation between the perturbation-series coefficients $r_l$ of an observable and the coefficients $f_k^{(p)}$ obtained by differentiating it as in eq. (8) is exactly invertible: eq. (10) expresses every $r_l$ as $f_l^{(p)}$ plus the same factorial factor $(2\beta_0/p)^l\,\Gamma(l+1+pb)/\Gamma(1+pb)$ that controls the known asymptotic growth in eq. (7), with the strength given by a weighted sum of the earlier $f_k^{(p)}$. Because eq. (10) is an equality rather than an asymptotic statement, eq. (11) turns the first $L$ known coefficients into a systematic approximation for all higher coefficients, valid at every order and not only for $l\gg1$. The series is then rewritten as a short renormalon-subtracted sum plus a factorially growing tail, and the tail is Borel-summed; the pole of the Borel integral produces a term of order $(\Lambda/Q)^p$, which is absorbed into the power correction. Improved scale stability is demonstrated for the static energy, the quark pole mass, and the polarized Bjorken sum rule, all with four known orders.

Load-bearing premise

The load-bearing premise, which the paper admits the manipulations in eqs. (8)-(10) do not prove, is that the first four known coefficients already sit in the factorial-growth regime, so that truncating the strength sum at $L=4$ still gives the correct high-order tail rather than being contaminated by subleading renormalons or by the next power correction.

Editorial extensions

If this is right

For any observable with four known coefficients, the factorial tail can be summed rather than set to zero, so the truncation uncertainty shrinks to the residual scale dependence of the subtracted series plus the Borel sum.
The pole-mass series, whose fixed-order version is unusable, becomes nearly as scale-stable as the static force after minimal renormalon subtraction, supporting sub-percent charm and bottom quark mass determinations.
For the static energy, the summed perturbation series separates cleanly from the linear power term, so fits to lattice data can distinguish the power correction that fixed-order perturbation theory cannot.
For the polarized Bjorken sum rule, summing the $p=2$ and $p=4$ factorial growth markedly stabilizes the scale variation, making a controlled $\alpha_s$ determination from deep-inelastic data plausible.
Factorial summation is at least as important as logarithmic resummation, because the renormalization-constrained factorial growth outruns a typical large-logarithm series.

Reading between the lines

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

If eq. (10) survives higher orders, the same construction should apply to any observable with an operator-product expansion, including cases with several power corrections whose anomalous dimensions make the exponents non-integer; the paper notes this case is cumbersome but not excluded.
A direct falsifier within reach is the five-loop coefficient of one of the three series: comparing it with the prediction of eq. (11) would show whether the first four terms already probe the asymptotic factorial regime.
The near equality of the factorial strengths extracted from the pole mass and the static energy points to a common infrared origin for the $p=1$ growth, which could be elevated from a numerical coincidence into a consistency test of the method.
One could apply the same summation with only two or three known coefficients and use the growth of the residual scale dependence as a diagnostic for how many terms are needed before the low-order regime sets in.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You know the background, so here is the short version. This is a conference proceedings summarizing the author's minimal renormalon subtraction (MRS) program from refs. [1–4]. What is genuinely new in this text is small: the preliminary Bjorken sum rule plots with multiple power corrections (p = 2, 4, 6), and even those are explicitly preliminary, with no formulas given. For the rest, the paper is an exposition of the author's earlier work, not a new result.

What it does well: the exact inversion leading to eq. (10) is clean and correct; the matrix structure of the renormalization-group constraint is useful; and the cancellation in Table 2 for the static energy is genuinely striking. The paper is also admirably honest: it repeatedly flags that the key step, eq. (11), is not proven but required by consistency with the known asymptotic behavior. The three applications use published, independent coefficients, and no parameters are fitted to the data shown. That is good practice.

The soft spot is exactly where the stress-test note lands. Eq. (10) is exact at all orders, but eq. (11) truncates the strength sum at L=4 and then predicts r_l for l>=4 using only that truncated strength. The paper concedes this is a consistency assumption, not a proof. There is no bound on the omitted terms, no error estimate on the predicted r_4, and the claim that eq. (11) is a systematic approximation valid at any order is the load-bearing assertion. Subleading renormalons and anomalous dimensions, especially in the Bjorken case, could contaminate the first few f_k. The matching of the pole mass and static energy strengths in Section 5.2 and the scale variation plots are suggestive, but they do not quantify the truncation error. The author knows this; the paper would be much stronger with one concrete test against a known five-loop coefficient in some quantity, or a derived bound on the tail.

Who should read this: lattice and perturbative QCD practitioners who want a compact introduction to MRS, and anyone working on alpha_s or quark-mass extractions. The underlying program deserves serious referee time. As a standalone research paper, this proceedings is more of a status report than a new contribution, but it is a fair and useful one. If I were the editor, I would send it to review with the request that the Bjorken multi-power formulas be included or the section clearly marked as a placeholder. The central conjecture remains unproven, but the paper is honest about that, and the evidence presented is enough to make it worth engaging with seriously.

Referee Report

3 major / 8 minor

Summary. This proceedings paper argues that renormalization-scale independence of a QCD observable with a single hard scale relates the factorial growth of perturbative coefficients to power corrections. It constructs F^(p)(Q) in eq. (8), derives the triangular relation eq. (9), and inverts it exactly to eq. (10), which expresses each r_l in terms of lower-order f_k. The paper then truncates this exact inversion at L known coefficients and proposes eqs. (11)-(12) as a prediction of the factorial tail for all l ≥ L. Adding the predicted tail back and Borel-summing it defines 'minimal renormalon subtraction' (MRS). The method is applied to the static energy, quark pole mass, and polarized Bjorken sum rule, with claims of improved scale stability and controlled truncation uncertainties for alpha_s and quark-mass determinations.

Significance. If the truncated tail prediction were proven, this would be a practically important and elegant result: the first four coefficients of a series would determine the dominant renormalon, which could then be subtracted and Borel-summed, removing a major source of truncation uncertainty in quantities such as alpha_s and quark masses. The exact linear algebra from eq. (8) to eq. (10) is clean, and the paper uses only published coefficients with no parameters fitted to the data shown. The cancellation in Table 2 and the matching of the factorial strengths of the static energy and pole mass in section 5.2 are striking numerical checks. The significance is conditional because the central step, eq. (11), rests on an unproven assumption about the growth of the f_l^(p) coefficients.

major comments (3)

[Section 3, eqs. (10)-(12)] Equation (11) is presented as 'a systematic approximation valid at any order,' but this is the load-bearing assumption and it is not proven. The exact identity eq. (10) contains the term f_l^(p) and the full sum over k < l; eqs. (11)-(12) drop f_l^(p) for all l ≥ L and truncate the strength sum at L-1. The paper concedes this in the paragraph after eq. (10): 'The manipulations in eqs. (8)-(10) do not prove this property, but consistency with the methods [14] yielding eq. (7) require it.' No bound on the omitted terms is given, and the first predicted coefficient, r_4 (or v_4), has no error estimate. Subleading renormalons or anomalous dimensions could contaminate f_4 and the tail. The numerical checks in section 5 are suggestive but do not bound the truncation error. Please either prove a bound, provide a diagnostic that measures the size of the omitted terms (for example by varying L), or state clearly that this is an assumption whose validity is being tested.
[Section 5.3, eq. (27)] The Bjorken sum-rule application relies on an additional assumption, not present in the static-energy and pole-mass cases, that the higher-twist contributions can be approximated as pure powers p=2,4,6. The text itself notes that 'the powers are not pure integers' and that once anomalous dimensions give the higher-twist terms a Q-dependence 'the treatment of more than one power correction becomes cumbersome.' No sensitivity study to this assumption is shown, and the figure is labeled 'Preliminary!'. Consequently the claim in Section 6 that truncation uncertainties can now be controlled does not yet apply to this application.
[Section 6] The conclusion that 'truncation uncertainties can now be controlled' goes beyond what is demonstrated by the scale-variation plots. The plots show three choices of s and no quantitative definition or estimate of the truncation error is given. For a paper whose stated promise is controlled truncation uncertainties, the author should either define a truncation-error estimator (e.g., the residual scale dependence after MRS, or the change between L=3 and L=4) and quote numbers for the three applications, or soften the conclusion.

minor comments (8)

[Abstract] In the abstract, 'three quantities four which' should read 'three quantities for which'.
[Section 3, after eq. (10)] The sentence 'Equation (10) holds at every order, hence already at low orders' should be reworded, because eq. (10) is exact for all orders but the subsequent truncated prediction eq. (11) is not shown to be exact at low orders.
[Section 4.2] In the sentence introducing eq. (16), 'two the two sums' should be 'the two sums'.
[Section 5.1] In the caption of Figure 3, 'Figure 3 shoes' should be 'Figure 3 shows'.
[Section 5.3] In the sentence 'the the powers are not pure integers', the duplicate 'the' should be removed.
[Section 6] In the outlook, 'As it it standard' should be 'As it is standard'.
[Section 5.1] The qualitative verdicts 'pretty good, great, horrible, and great again' would be more useful as quantitative measures of scale variation, e.g., the maximum spread across s.
[Section 5.3] The generalized version of eqs. (11), (12), and (20) for multiple power corrections is not displayed; since Fig. 5 uses it, a formula or an explicit pointer to the derivation in ref. [1] is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No by-construction circularity: the RG inversion eq. (10) is exact algebra and the strength eq. (12) is computed, not fitted. The main caveats are heavy self-citation and in-sample validation, which are correctness concerns, not circular reductions.

full rationale

The derivation chain eqs. (8)-(10) is an algebraic inversion of the renormalization-group constraint; eq. (10) follows by row-by-row solution of a lower-triangular system and contains no fitted parameter. The factorial-growth template eq. (7) is imported from the external review [14], and the strength in eq. (12) is computed from the same low-order coefficients, not obtained by fitting to the quantities whose convergence is then displayed. The applications to static energy, pole mass, and Bjorken sum rule use independent published coefficients. The genuinely weak point—that the truncation at L=4 and the claim that eq. (11) is a systematic approximation at any order rest on an admitted consistency assumption ('The manipulations in eqs. (8)-(10) do not prove this property, but consistency with the methods [14] yielding eq. (7) require it')—is a rigor risk, not a circularity: the paper does not define the tail in terms of the predicted quantity, nor does it fit a parameter and rename it a prediction. The remaining caveats are that the model is validated on the same series used to construct it and that the central new-validity claim is attributed to the author's own ref. [1]; neither reduces the results to their inputs by construction, so the score is low.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The method introduces no new particles or forces. Its free parameters are modeling choices (p and L) rather than fits to data; the load-bearing assumptions are the factorization ansatz, the log-only Q dependence, the unproved all-orders tail, and the Borel prescription.

free parameters (2)

power p of the leading power correction = 1 for static energy and pole mass; 2, 4, 6 for Bjorken sum rule
Not fitted to the data shown; p is assigned from OPE and renormalon arguments, but all summed results depend on this assignment, and the wrong p would change the tail.
number of known coefficients L = 4
Set by the availability of published coefficients; the strength R0^(p) in eq. (12) truncates the sum at L, so the tail prediction depends on L. This is a modeling choice, not a fit.

assumptions (6)

domain assumption Physical observables obey the factorization R = r_-1 + R_pert + C_p Lambda^p / Q^p (eq. 1).
Starting point for the method; justified by OPE and EFT in the text, but not derived here.
domain assumption The only Q dependence of R_pert in massless QCD enters through ln(mu/Q) in the coefficients r_l.
Section 3 states this is 'remarkably and crucially' the case; it underlies the derivative construction in eq. (8).
ad hoc to paper The factorial growth of r_l is determined at every order by the low-order f_k through RG consistency, so the tail in eq. (11) is valid for l >= L.
The text says 'The manipulations in eqs. (8)-(10) do not prove this property, but consistency with the methods [14] require it.' This is the unproved leap.
domain assumption Borel summation by principal value, with the contour ambiguity absorbed into C_p, is the correct resummation.
Section 4.2 chooses principal value and moves the ambiguity into the power correction; this fixes the prescription but is not derived.
domain assumption A geometric scheme for alpha_s exists and eqs. (9)-(14) are valid there; results are scheme-independent after switching back.
Footnote 2 defers details to ref. [1]; the transformation is not shown in this paper.
ad hoc to paper Higher-twist corrections for the Bjorken sum rule can be approximated as pure powers p = 2, 4, 6 without anomalous-dimension complications.
Section 5.3 notes fractional dimensions make multiple powers cumbersome and the p = (2,4,6) sum does not help much.

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Cite this review

Pith. "Pith review of Factorial growth in perturbation theory, power corrections: precise extraction of quark masses and $\alpha_\text{s}$." pith.science (2026). https://pith.science/paper/X7UQXHSO

@misc{pith2026250520531,
  author       = {Pith},
  title        = {Pith review of: Factorial growth in perturbation theory, power corrections: precise extraction of quark masses and $\alpha_\texts$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7UQXHSO}},
  note         = {Machine review of arXiv:2505.20531}
}

abstract

These proceedings summarize a newly found connection between the factorial growth of coefficients in perturbative QCD and power corrections to the perturbation series, discussed in refs. [1-4]. The improved convergence is shown for three quantities four which four terms in the series are available: the static energy, the quark pole mass, and the polarized Bjorken sum rule. Prospects for determinations of $\alpha_\text{s}$ with controlled truncation uncertainties are discussed, as was found earlier in quark-mass determinations [3,5].

Figures

Figures reproduced from arXiv: 2505.20531 by the authors.

**Figure 1.** Comparison of factorially growing terms in the perturbative series coefficients with a typical (if modest) “large” logarithm. It seems at least as important to sum factorials (cf. section 4.2) as logs. 𝛽0 and 𝑏 are taken for 𝑛 𝑓 = 3. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** s=1 s=2 Scale variation in the 𝛼2 scheme. Top: 𝑅˜(𝑞) and 𝐹(1/𝑟); neither suffers the 𝑝 = 1 growth. Bottom: 𝑅(1/𝑟) (with 𝑝 = 1 growth) and 𝑅MRS(1/𝑟) (after MRS). Red, green, and blue curves correspond to 𝑠 = 1 2 , 𝑠 = 1, and 𝑠 = 2, respectively. Solid (dashed) curves correspond to a running (fixed) 𝛼s in the ultrasoft ln 𝛼s . Note that the vertical scale for 𝑅(1/𝑟) is twice that of the other three plots. From ref. [1… view at source ↗

**Figure 3.** s=1 s=2 Scale variation in the 𝛼2 scheme of the Borel sum 𝑅B(1/𝑟) (left) and the 𝐿 = 4-subtracted series 𝑅RS(1/𝑟) (right). Curve and color code as in figure 2. From ref. [1]. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Left: fixed-order perturbation theory for 𝑅 s=1 s=2 (𝑚¯ ) ≡ 𝑀/𝑚¯ − 1 vs. Λ/𝑚¯ for 𝜇 = 𝑠𝑚¯ ; right: factorially summed 𝑅MRS(𝑚¯ ). In both, the MS scheme is used for 𝛼s , and 𝑠 ∈ { 1 2 , 1, 2} with the same color code as in figure 2. From ref. [4]. 10 [PITH_FULL_IMAGE:f…

**Figure 5.** Figure 5: Preliminary! Top left: fixed-order perturbation theory for 𝑅(𝑄) in eq. (29); top right: MRS factorial sum for growth with 𝑝 = 2; bottom left: MRS factorial sum for growth with (𝑝1, 𝑝2) = (2, 4); bottom right: MRS factorial sum for growth with (𝑝1, 𝑝2, 𝑝3) = (2, 4, 6). …

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Works this paper leans on

63 extracted references · 30 canonical work pages · cited by 1 Pith paper

[14]

Beneke,Renormalons,Phys

M. Beneke,Renormalons,Phys. Rept.317 (1999) 1 [hep-ph/9807443]

arXiv 1999
[1]

Kronfeld,Factorial growth at low orders in perturbative QCD: control over truncation uncertainties,J

A.S. Kronfeld,Factorial growth at low orders in perturbative QCD: control over truncation uncertainties,J. High Energy Phys.12(2023) 108 [2310.15137]

arXiv 2023
[2]

Komĳani,A discussion on leading renormalon in the pole mass, J

J. Komĳani,A discussion on leading renormalon in the pole mass, J. High Energy Phys. 2017 (2017) 62 [1701.00347]

arXiv 2017
[3]

Brambilla, J

TUMQCD: N. Brambilla, J. Komĳani, A.S. Kronfeld and A. Vairo,Relations between heavy-light meson and quark masses, Phys. Rev. D97(2018) 034503 [1712.04983]

arXiv 2018
[4]

More on minimal renormalon subtraction

A.S. Kronfeld,More on minimal renormalon subtraction, PoS LATTICE2023(2024) 341 [2401.07380]

work page Pith review arXiv 2024
[5]

Bazavov et al.,Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys

Fermilab Lattice, MILC, TUMQCD: A. Bazavov et al.,Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys. Rev. D98(2018) 054517 [1802.04248]

arXiv 2018
[6]

Navas et al.,Review of particle physics,Phys

Particle Data Group: S. Navas et al.,Review of particle physics,Phys. Rev. D110 (2024) 030001

work page 2024
[7]

Aoki et al.,FLAG Review 2024, 2411.04268

Flavour Lattice A veraging Group: Y. Aoki et al.,FLAG Review 2024, 2411.04268

arXiv 2024

Show all 63 references

[8]

Wilson,The renormalization group: Critical phenomena and the Kondo problem, Rev

K.G. Wilson,The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys.47(1975) 773

1975
[9]

Bender and T.T

C.M. Bender and T.T. Wu,Large-order behavior of perturbation theory, Phys. Rev. Lett.27 (1971) 461

1971
[10]

Bender and T.T

C.M. Bender and T.T. Wu,Anharmonic oscillator II: A study of perturbation theory in large order,Phys. Rev. D7 (1973) 1620

1973
[11]

Gross and A

D.J. Gross and A. Neveu,Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D10(1974) 3235

1974
[12]

Lautrup,On high order estimates in QED, Phys

B.E. Lautrup,On high order estimates in QED, Phys. Lett. B69 (1977) 109

1977
[13]

Duncan,The Conceptual Framework of Quantum Field Theory, Oxford University Press, Oxford (2012), 10.1093/acprof:oso/9780199573264.001.0001

A. Duncan,The Conceptual Framework of Quantum Field Theory, Oxford University Press, Oxford (2012), 10.1093/acprof:oso/9780199573264.001.0001

2012
[15]

Lee,Renormalons beyond one loop, Phys

T. Lee,Renormalons beyond one loop, Phys. Rev. D56 (1997) 1091 [hep-th/9611010]

1997 arXiv
[16]

Lee,Normalization constants of large order behavior, Phys

T. Lee,Normalization constants of large order behavior, Phys. Lett. B462(1999) 1 [hep-ph/9908225]

1999 arXiv
[17]

Pineda,Determination of the bottom quark mass from theΥ(1S) system, J

A. Pineda,Determination of the bottom quark mass from theΥ(1S) system, J. High Energy Phys. 06(2001) 022 [hep-ph/0105008]. 13 Factorial growth Andreas S. Kronfeld

2001 arXiv
[18]

Hoang, A

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 [0803.4214]

2008 arXiv
[19]

Brown, L.G

L.S. Brown, L.G. Yaffe and C.-X. Zhai,Large-order perturbation theory for the electromagnetic current-current correlation function, Phys. Rev. D46(1992) 4712 [hep-ph/9205213]

1992 arXiv
[20]

Gross and F

D.J. Gross and F. Wilczek,Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30(1973) 1343

1973
[21]

Politzer,Reliable perturbative results for strong interactions?, Phys

H.D. Politzer,Reliable perturbative results for strong interactions?, Phys. Rev. Lett.30 (1973) 1346

1973
[22]

Jones,Two-loop diagrams in Yang-Mills theory, Nucl

D.R.T. Jones,Two-loop diagrams in Yang-Mills theory, Nucl. Phys. B75 (1974) 531

1974
[23]

Caswell,Asymptotic behavior of non-Abelian gauge theories to two-loop order, Phys

W.E. Caswell,Asymptotic behavior of non-Abelian gauge theories to two-loop order, Phys. Rev. Lett.33(1974) 244

1974
[24]

Tarasov, A.A

O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov,The Gell-Mann–Low function of QCD in the three-loop approximation,Phys. Lett. B93(1980) 429

1980
[25]

Larin and J.A.M

S.A. Larin and J.A.M. Vermaseren,The three-loop QCD𝛽-function and anomalous dimensions, Phys. Lett. B303 (1993) 334 [hep-ph/9302208]

1993 arXiv
[26]

van Ritbergen, J.A.M

T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin,The four-loop𝛽-function in quantum chromodynamics,Phys. Lett. B400 (1997) 379 [hep-ph/9701390]

1997 arXiv
[27]

Czakon,The four-loop QCD𝛽-function and anomalous dimensions, Nucl

M. Czakon,The four-loop QCD𝛽-function and anomalous dimensions, Nucl. Phys. B710 (2005) 485 [hep-ph/0411261]

2005 arXiv
[28]

Zoller,Four-loop QCD𝛽-function with different fermion representations of the gauge group,J

M.F. Zoller,Four-loop QCD𝛽-function with different fermion representations of the gauge group,J. High Energy Phys.2016(2016) 118 [1608.08982]

2016 arXiv
[29]

Baikov, K.G

P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,Five-loop running of the QCD coupling constant,Phys. Rev. Lett.118 (2017) 082002 [1606.08659]

2017 arXiv
[30]

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, J. High Energy Phys.2017 (2017) 090 [1701.01404]

2017 arXiv
[31]

Luthe, A

T. Luthe, A. Maier, P. Marquard and Y. Schröder,Complete renormalization of QCD at five loops, J. High Energy Phys.2017 (2017) 020 [1701.07068]

2017 arXiv
[32]

Appelquist, M

T. Appelquist, M. Dine and I.J. Muzinich,Static limit of quantum chromodynamics,Phys. Rev. D17(1978) 2074

1978
[33]

Brambilla, A

N. Brambilla, A. Pineda, J. Soto and A. Vairo,Infrared behavior of the static potential in perturbative QCD,Phys. Rev. D60 (1999) 091502 [hep-ph/9903355]

1999 arXiv
[34]

Brambilla, A

N. Brambilla, A. Pineda, J. Soto and A. Vairo,Potential NRQCD: An effective theory for heavy quarkonium, Nucl. Phys. B566 (2000) 275 [hep-ph/9907240]. 14 Factorial growth Andreas S. Kronfeld

2000 arXiv
[35]

Fischler,Quark-antiquark potential in QCD,Nucl

W. Fischler,Quark-antiquark potential in QCD,Nucl. Phys. B129 (1977) 157

1977
[36]

Billoire,How heavy must quarks be in order to build Coulombic𝑞 ¯𝑞 bound states?,Phys

A. Billoire,How heavy must quarks be in order to build Coulombic𝑞 ¯𝑞 bound states?,Phys. Lett. B92 (1980) 343

1980
[37]

Schröder,The static potential in QCD to two loops, Phys

Y. Schröder,The static potential in QCD to two loops, Phys. Lett. B447 (1999) 321 [hep-ph/9812205]

1999 arXiv
[38]

Kniehl, A.A

B.A. Kniehl, A.A. Penin, M. Steinhauser and V.A. Smirnov,Non-Abelian𝛼3 𝑠(𝑠)/(𝑚𝑞𝑟2) heavy-quark–antiquark potential, Phys. Rev. D65(2002) 091503 [hep-ph/0106135]

2002 arXiv
[39]

Smirnov, V.A

A.V. Smirnov, V.A. Smirnov and M. Steinhauser,Fermionic contributions to the three-loop static potential,Phys. Lett. B668 (2008) 293 [0809.1927]

2008 arXiv
[40]

Anzai, Y

C. Anzai, Y. Kiyo and Y. Sumino,Static QCD potential at three-loop order, Phys. Rev. Lett. 104 (2010) 112003 [0911.4335]

2010 arXiv
[41]

Smirnov, V.A

A.V. Smirnov, V.A. Smirnov and M. Steinhauser,Three-loop static potential,Phys. Rev. Lett. 104 (2010) 112002 [0911.4742]

2010 arXiv
[42]

Lee, A.V

R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser,Analytic three-loop static potential, Phys. Rev. D94 (2016) 054029 [1608.02603]

2016 arXiv
[43]

Brambilla, X

N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo,The logarithmic contribution to the QCD static energy at N4LO, Phys. Lett. B647 (2007) 185 [hep-ph/0610143]

2007 arXiv
[44]

Leino, A

TUMQCD: V. Leino, A. Bazavov, N. Brambilla, A.S. Kronfeld, J. Mayer-Steudte, P. Petreczky et al.,Strong coupling in (2+1+1)-flavor QCD,PoS LATTICE2024(2025) 298 [2502.01453]

2025 arXiv
[45]

Brambilla, R.L

TUMQCD: N. Brambilla, R.L. Delgado, A.S. Kronfeld, V. Leino, P. Petreczky, S. Steinbeißer et al.,Static energy in (2+ 1+ 1)-flavor lattice QCD: Scale setting and charm effects,Phys. Rev. D107 (2023) 074503 [2206.03156]

2023 arXiv
[46]

Bazavov et al.,Lattice QCD ensembles with four flavors of highly improved staggered quarks, Phys

MILC: A. Bazavov et al.,Lattice QCD ensembles with four flavors of highly improved staggered quarks, Phys. Rev. D87(2013) 054505 [1212.4768]

2013 arXiv
[47]

Fermilab Lattice, MILC:A.Bazavovetal., 𝐵-and 𝐷-mesonleptonicdecayconstantsfrom four-flavor lattice QCD,Phys. Rev. D98(2018) 074512 [1712.09262]

2018 arXiv
[48]

Gray, D.J

N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher,Three loop relation of quarkMS and pole masses, Z. Phys. C48(1990) 673

1990
[49]

Chetyrkin and M

K.G. Chetyrkin and M. Steinhauser,Short distance mass of a heavy quark at order𝛼3 𝑠, Phys. Rev. Lett.83(1999) 4001 [hep-ph/9907509]

1999 arXiv
[50]

Chetyrkin and M

K.G. Chetyrkin and M. Steinhauser,The relation between theMS and the on-shell quark mass at order𝛼3 s,Nucl. Phys. B573 (2000) 617 [hep-ph/9911434]. 15 Factorial growth Andreas S. Kronfeld

2000 arXiv
[51]

Melnikov and T

K. Melnikov and T. van Ritbergen,The three loop relation between theMSand the pole quark masses, Phys. Lett. B482 (2000) 99 [hep-ph/9912391]

2000 arXiv
[52]

Marquard, A.V

P. Marquard, A.V. Smirnov, V.A. Smirnov and M. Steinhauser,Quark mass relations to four-loop order in perturbative QCD, Phys. Rev. Lett.114 (2015) 142002 [1502.01030]

2015 arXiv
[53]

Marquard, A.V

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. D 94(2016) 074025 [1606.06754]

2016 arXiv
[54]

Bjorken,Applications of the chiral U(6)× U(6) algebra of current densities,Phys

J.D. Bjorken,Applications of the chiral U(6)× U(6) algebra of current densities,Phys. Rev. 148 (1966) 1467

1966
[55]

Bjorken,Inelastic scattering of polarized leptons from polarized nucleons, Phys

J.D. Bjorken,Inelastic scattering of polarized leptons from polarized nucleons, Phys. Rev. D 1 (1970) 1376

1970
[56]

Kodaira, S

J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu,QCD effects in polarized electroproduction, Phys. Rev. D20(1979) 627

1979
[57]

Kodaira, S

J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu,QCD higher order effects in spin dependent deep inelastic electroproduction, Nucl. Phys. B159 (1979) 99

1979
[58]

Blümlein,The theory of deeply inelastic scattering, Prog

J. Blümlein,The theory of deeply inelastic scattering, Prog. Part. Nucl. Phys.69(2013) 28 [1208.6087]

2013 arXiv
[59]

Gorishnii, S.A

S.G. Gorishnii, S.A. Larin, F.V. Tkachov and K.G. Chetyrkin,Higher QCD corrections to the Bjorken sum rule, Phys. Lett. B137 (1984) 230

1984
[60]

Larin, F.V

S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren,𝛼3 𝑠 correction to the Bjorken sum rule, Phys. Rev. Lett.66(1991) 862

1991
[61]

Baikov, K.G

P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,Adler function, Bjorken sum rule, and the Crewther relation to order𝛼4 𝑠 in a general gauge theory, Phys. Rev. Lett.104 (2010) 132004 [1001.3606]

2010 arXiv
[62]

K.G.Chetyrkin, Adlerfunction, Bjorken SumRule andCrewther-Broadhurst-Kataev relation with generic fermion representations at order𝑂(𝛼4 𝑠),Nucl. Phys. B985(2022) 115988 [2206.12948]

2022 arXiv
[63]

Kutz, J.R

T. Kutz, J.R. Pybus, D.W. Upton, C. Cotton, A. Deshpande, A. Deur et al.,High precision measurements of𝛼𝑠 at the future EIC, Phys. Rev. D110 (2024) 074004 [2406.05591]. 16

2024 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

Beneke,Renormalons,Phys

M. Beneke,Renormalons,Phys. Rept.317 (1999) 1 [hep-ph/9807443]

arXiv 1999

[2] [1]

Kronfeld,Factorial growth at low orders in perturbative QCD: control over truncation uncertainties,J

A.S. Kronfeld,Factorial growth at low orders in perturbative QCD: control over truncation uncertainties,J. High Energy Phys.12(2023) 108 [2310.15137]

arXiv 2023

[3] [2]

Komĳani,A discussion on leading renormalon in the pole mass, J

J. Komĳani,A discussion on leading renormalon in the pole mass, J. High Energy Phys. 2017 (2017) 62 [1701.00347]

arXiv 2017

[4] [3]

Brambilla, J

TUMQCD: N. Brambilla, J. Komĳani, A.S. Kronfeld and A. Vairo,Relations between heavy-light meson and quark masses, Phys. Rev. D97(2018) 034503 [1712.04983]

arXiv 2018

[5] [4]

More on minimal renormalon subtraction

A.S. Kronfeld,More on minimal renormalon subtraction, PoS LATTICE2023(2024) 341 [2401.07380]

work page Pith review arXiv 2024

[6] [5]

Bazavov et al.,Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys

Fermilab Lattice, MILC, TUMQCD: A. Bazavov et al.,Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys. Rev. D98(2018) 054517 [1802.04248]

arXiv 2018

[7] [6]

Navas et al.,Review of particle physics,Phys

Particle Data Group: S. Navas et al.,Review of particle physics,Phys. Rev. D110 (2024) 030001

work page 2024

[8] [7]

Aoki et al.,FLAG Review 2024, 2411.04268

Flavour Lattice A veraging Group: Y. Aoki et al.,FLAG Review 2024, 2411.04268

arXiv 2024