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REVIEW 2 major objections 4 minor 94 references

Four loop renormalization of QCD in the Curci-Ferrari gauge

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Four-loop QCD beta function reproduced in Curci-Ferrari gauge

desk verdict Solid four-loop Curci-Ferrari computation, but Eq. (4.5) inverts the gauge-parameter conversion and blocks verification of the nonzero-α five-loop mMOM results. read the letter →

arxiv 2412.20950 v2 pith:KCUJCEPS submitted 2024-12-30 hep-th

J.A. Gracey This is my paper · ORCID
classification hep-th MSC 81T1581T1881T1381V05 PACS 11.15.-q12.38.-t12.38.Bx
keywords Curci-FerrarigaugeQCDbetafunctionfour-looprenormalizationSlavnov-TayloridentityminimalmomentumsubtractionschemequarkmassanomalousdimensionBRSTinvariantoperatormultiloopFeynmanintegrals
topics Yang-Mills Mass Gap
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

This paper renormalizes Quantum Chromodynamics in the non-linear Curci-Ferrari gauge to four loops in the MSbar scheme. Its central result is that the four-loop MSbar beta function is reproduced from a Slavnov-Taylor identity alone, without evaluating any vertex function. The same identity yields the four-loop anomalous dimensions of the fields, the gauge parameter, and the BRST-invariant dimension-two gluon and ghost mass operator. Because the minimal momentum subtraction (mMOM) scheme inherits the identity, the paper derives the five-loop mMOM beta function and quark mass anomalous dimension as corollaries. If correct, these are new results accurate to the stated loop orders.

What carries the argument

The load-bearing object is the Slavnov-Taylor identity $Z_g = Z_\alpha^2 / \sqrt{Z_A Z_c}$ of the Curci-Ferrari gauge, which connects the coupling-constant renormalization to the gluon, ghost, and gauge-parameter renormalization constants. It is the nonlinear-gauge generalization of Taylor's Landau-gauge non-renormalization theorem, and it lets the author extract $Z_g$ from two-point functions alone. A second identity, $Z_O = Z_\alpha^2 / (Z_A Z_c)$, fixes the anomalous dimension of the BRST-invariant dimension-two mass operator. The mMOM scheme is defined by demanding that the same combination of renormalization constants equals unity, so the conversion from MS to mMOM is built directly on these identities.

What would settle it

An independent direct evaluation of the ghost-gluon vertex in the Curci-Ferrari gauge at four loops, giving a coupling renormalization constant different from the Slavnov-Taylor value, would refute the central claim; likewise, any future five-loop mMOM computation that disagrees with the published beta function or quark mass anomalous dimension would falsify the corollary.

Watch

Extended reading notes

Core claim

The paper establishes that the Curci-Ferrari gauge, despite its extra quartic ghost interaction and non-trivial gauge-parameter renormalization, provides a more economical route to high-order QCD renormalization group functions. By computing only the gluon, ghost, and quark two-point functions to four loops, and using the Slavnov-Taylor identity $Z_g = Z_\alpha^2 / \sqrt{Z_A Z_c}$, the author reproduces the known four-loop MSbar $\beta$ function for nonzero gauge parameter. A companion identity fixes the renormalization of the BRST-invariant dimension-two mass operator from the same two-point functions. The scheme is then converted to mMOM, producing four-loop field anomalous dimensions and, via renormalization-group consistency, five-loop $\beta$ function and quark mass anomalous dimension in that scheme.

Load-bearing premise

The argument assumes the Slavnov-Taylor identity $Z_g = Z_\alpha^2 / \sqrt{Z_A Z_c}$ holds to all orders in the Curci-Ferrari gauge; the paper verifies it at four loops, but if a subtlety in the renormalization of the quartic ghost vertex broke the identity at this order, the $\beta$ function and five-loop mMOM results would not follow.

Editorial extensions

If this is right

  • The four-loop field and gauge-parameter anomalous dimensions in the Curci-Ferrari gauge are new results for arbitrary colour group and gauge parameter.
  • The five-loop mMOM beta function and quark mass anomalous dimension are new and can update renormalization-flow analyses of the strong interaction.
  • The method avoids vertex-function computation, cutting the number of required graphs compared to the linear covariant gauge approach for the same output.
  • The structure of the Curci-Ferrari gauge at $\alpha = 1$ removes the dipole term of the gluon propagator, offering a practical route toward a six-loop MSbar beta function.

Reading between the lines

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

  • The absence of even zeta values in the five-loop mMOM functions for nonzero gauge parameter may be a general property of momentum-subtraction schemes with no $O(a)$ corrections, not tied to the Curci-Ferrari gauge.
  • The same identity-based strategy could be applied to other observables, such as the anomalous dimensions of gluonic twist-2 operators, where the $\alpha = 1$ simplification would reduce the integration burden.
  • If the Slavnov-Taylor identity extends to the massive Curci-Ferrari model, the approach could also produce high-loop results for the massive theory that is used to model infrared gluon dynamics.
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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

2 major / 4 minor

Summary. The paper reports a four-loop renormalization of massless QCD in the nonlinear Curci-Ferrari gauge in the MSbar scheme. The author computes the gluon, ghost, quark, and gauge-parameter renormalization constants using the Forcer package; verifies the Slavnov-Taylor identities relating the coupling and the dimension-two mass operator to the field renormalizations; reproduces the known four-loop MSbar beta function without evaluating a vertex function; and constructs mMOM-scheme field and gauge-parameter anomalous dimensions to four loops. A renormalization-group conversion is then used to derive five-loop mMOM beta function and quark-mass anomalous dimension. The paper also discusses the prospects for computing a six-loop MSbar beta function in this gauge.

Significance. If the results are correct, this is a substantial computational advance. The four-loop MSbar anomalous dimensions are new, the ST-identity route to the beta function is elegant, and the five-loop mMOM quantities are genuinely new. The paper has several strong internal checks: reproduction of the known four-loop MSbar beta function, explicit verification of the ST identities to four loops, agreement with the three-loop Curci-Ferrari results of [31], and agreement with known Landau-gauge and mMOM limits at alpha=0. The ancillary data also support reproducibility. However, key displayed identities and the gauge-parameter conversion contain reciprocal errors that currently block verification of the five-loop corollaries.

major comments (2)
  1. [Sec. 2, Eqs. (2.10), (2.12), (2.13)] The printed Slavnov-Taylor identities for the mass operator are inconsistent with the rest of the paper. From (2.11), Z_g = Z_alpha^2 / sqrt(Z_A Z_c), and from (2.12), which the paper says is checked explicitly, gamma_O = gamma_A + gamma_c - 2(gamma_A + gamma_alpha). The latter identity corresponds to Z_O = Z_A Z_c / Z_alpha^2, not the printed Z_O = Z_alpha^2 / (Z_A Z_c). Consequently the combination of (2.10) and (2.11) should give Z_O = Z_alpha^2 / Z_g^2, not Z_O = Z_c Z_g^2 / Z_alpha^2 as printed in (2.13). Since Section 3 states that Z_g is extracted using (2.13), the derivation as printed is not reproducible. Please correct these relations and confirm that the reported four-loop checks were performed with the corrected identities.
  2. [Sec. 4, Eq. (4.5)] The gauge-parameter conversion in (4.5) is inverted relative to the paper's own convention (2.5). Equating bare gauge parameters gives alpha_mMOM = (Z_A^MS Z_alpha^mMOM) / (Z_alpha^MS Z_A^mMOM) alpha_MS; in the linear-gauge limit Z_alpha = 1 this reduces to alpha_mMOM = Z_A^MS / Z_A^mMOM alpha_MS, the reciprocal of the printed relation alpha_mMOM = Z_A^mMOM / Z_A^MS alpha_MS. This mapping is used in (4.4)-(4.7) to derive the five-loop mMOM beta function (4.8), the quark-mass anomalous dimension (4.9), and Appendix B. The alpha=0 comparisons with [58] cannot detect an error in the alpha-dependent conversion, and no nonzero-alpha cross-check is provided. As printed, the five-loop corollaries are not reproducible. Please correct the conversion and either re-derive the five-loop results with the corrected mapping or provide a direct four-loop cross-check at nonzero alpha.
minor comments (4)
  1. [Table 3] For the linear gauge, the six-loop total 19752679 does not equal the sum of the three entries 3375438 + 857240 + 16215639 = 20448317; please check the arithmetic or clarify what the total includes.
  2. [Sec. 4, after Eq. (4.3)] The sentence 'the four loop mMOM beta-function will not agree with the corresponding linear covariant gauge result' is slightly misleading because the two do agree at alpha=0; consider adding 'for alpha != 0'.
  3. [Sec. 4, around Eq. (4.2)] The statement that 'no coupling constant mapping is actually required' is strong given that the subsequent five-loop construction does require the mapping in (4.4)-(4.7); please clarify the distinction.
  4. [Throughout] There are several typographical errors, e.g., 'synomymous' in Section 1, 'vaccum' and 'intgrals' in Section 5, and 'bête-noire' in Section 6; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the four-loop MS results are independent computations checked against known beta, and the five-loop mMOM functions are standard RG conversions from known MS inputs.

full rationale

The derivation chain is self-contained against external benchmarks. The Slavnov-Taylor identities (2.10)-(2.13) are taken from Refs. [29,30], which are not authored by the present author, and the paper explicitly checks them at four loops by comparing the directly computed Z_O and Z_g with the known four-loop MS beta function; the reproduction of the known beta is a check, not an output that defines the inputs. The mMOM scheme is defined by the condition (4.2), with the mMOM field, ghost, and gauge-parameter renormalization constants fixed by the no-O(a) subtraction condition, so no parameter is fitted to a target quantity and then renamed as a prediction. The five-loop mMOM beta function and quark-mass anomalous dimension are obtained by the standard renormalization-group conversion formulas (4.6)-(4.7) from the known five-loop MS results, which are external inputs; this is a scheme transformation, not a circular derivation. The self-citations that appear ([31], [51], [59,60]) are used as lower-loop checks or contextual comparisons and are not load-bearing for the central claim. The apparent inversion in Eq. (4.5) noted in the skeptic brief is a potential algebraic consistency or correctness issue, not a circularity: the five-loop mMOM result would still be derived from known MS inputs, albeit possibly incorrectly if the printed mapping is implemented literally.

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

The paper introduces no free parameters and no new entities. It relies on established axioms: the renormalizability and BRST invariance of the Curci-Ferrari gauge, the Slavnov-Taylor identities, the correctness of the Forcer algorithm, the known five-loop MS beta function, and the standard RG conversion formalism.

assumptions (6)
  • domain assumption The Curci-Ferrari gauge Lagrangian (2.1) is renormalizable and BRST invariant.
    Standard property of the model from [16], used as the starting point for the whole calculation.
  • domain assumption The Slavnov-Taylor identities (2.10) and (2.11) from [29,30] hold to all orders in the Curci-Ferrari gauge.
    These proven identities are the backbone of the paper; they connect the mass operator and coupling renormalization constants to the field and gauge parameter renormalization constants. The paper verifies them to four loops but assumes their exact all-orders validity.
  • domain assumption The Forcer algorithm correctly computes the epsilon expansions of all required four-loop massless two-point integrals.
    The entire computation rests on this automated tool; its correctness is checked indirectly via the Slavnov-Taylor identities and the known beta function.
  • domain assumption The five-loop MS beta function of QCD is gauge-parameter independent and is known from [38,39,40,41].
    Used as an input in the renormalization group conversion to derive the five-loop mMOM beta function.
  • standard math The renormalization group conversion functions (4.4)-(4.7) are valid for the Curci-Ferrari gauge mMOM scheme.
    Standard RG scheme-change formalism; assumes the massless scheme definitions are consistent.
  • domain assumption The mMOM scheme is defined by requiring the renormalized ghost, gluon, and quark two-point functions have no O(a) corrections.
    This is the scheme definition from (4.2) and the surrounding text; it determines the field renormalization constants in the mMOM scheme.

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Pith. "Pith review of Four loop renormalization of QCD in the Curci-Ferrari gauge." pith.science (2026). https://pith.science/paper/KCUJCEPS

@misc{pith2026241220950,
  author       = {Pith},
  title        = {Pith review of: Four loop renormalization of QCD in the Curci-Ferrari gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCUJCEPS}},
  note         = {Machine review of arXiv:2412.20950}
}
read the original abstract

We renormalize Quantum Chromodynamics (QCD) when gauge fixed in the nonlinear Curci-Ferrari gauge to four loops in the modified minimal subtraction (MSbar) scheme. We reproduce the four loop QCD MSbar beta-function from the Slavnov-Taylor identity for this gauge which relates the coupling constant renormalization to the gluon, Faddeev-Popov ghost and gauge parameter anomalous dimensions. This is carried out for a nonzero gauge parameter, without having to evaluate a vertex function. The anomalous dimension of the BRST invariant dimension two gluon and ghost mass term is deduced from a similar Slavnov-Taylor identity for this gauge. Consequently we construct the renormalization group functions in the minimal momentum subtraction scheme to four loops. As a corollary we deduce the five loop beta-function and quark mass anomalous dimensions in the same scheme. We also outline the pros and cons of employing the Curci-Ferrari gauge to access the six loop QCD beta-function in the MSbar scheme.

Figures

Figures reproduced from arXiv: 2412.20950 by the authors.

Figure 1
Figure 1. Graphical structure of Gab O (p). We close this section by noting we used the Forcer package of [44, 45] to carry out the task of evaluating the relevant Green’s functions to renormalize (2.1). The Forcer algorithm computes the ϵ expansion in d = 4 − 2ϵ dimensions of massless 2-point Feynman integrals up to and including four loops. It is therefore ideally suited to the task and is written in the highly efficient sy… view at source ↗
Figure 2
Figure 2. Six loop graph contributing to gluon 2-point function. [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Six loop graph contributing to ghost 2-point function. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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Works this paper leans on

94 extracted references · 79 canonical work pages

  1. [31]

    Gracey, Phys

    J.A. Gracey, Phys. Lett. B552 (2003), 101

    work page 2003

  2. [58]

    Nogueira, J

    P. Nogueira, J. Comput. Phys. 105 (1993), 279

    work page 1993

  3. [1]

    Cucchieri & T

    A. Cucchieri & T. Mendes, PoS LA TTICE2007(2007), 297

    2007

  4. [2]

    Bogolubsky, E.M

    I.L. Bogolubsky, E.M. Ilgenfritz, M. M¨ uller-Preussker & A. Sternbeck, PoS LA TTICE2007 (2007), 290. 31

    2007

  5. [3]

    Sternbeck, L

    A. Sternbeck, L. von Smekal, D.B. Leinweber & A.G. Williams, PoS LA TTICE2007(2007), 340

    2007

  6. [4]

    Bogolubsky, E.M

    I.L. Bogolubsky, E.M. Ilgenfritz, M. M¨ uller-Preussker & A. Sternbeck, Phys. Lett. B676 (2009), 69

    2009

  7. [5]

    Cucchieri & T

    A. Cucchieri & T. Mendes, Phys. Rev. Lett. 100 (2008), 241601

    2008

  8. [6]

    Cucchieri & T

    A. Cucchieri & T. Mendes, Phys. Rev. D78 (2008), 094503

    2008

Show all 94 references
  1. [7]

    Oliveira & P.J

    O. Oliveira & P.J. Silva, Phys. Rev. D79 (2009), 031501

    2009

  2. [8]

    Boucaud, J.P

    Ph. Boucaud, J.P. Leroy, A.L. Yaounac, J. Micheli, O. P` ene & J. Rodr ´ ıguez-Quintero, J. High Energy Phys. 06 (2008), 099

    2008

  3. [9]

    Cornwall, Phys

    J.M. Cornwall, Phys. Rev. D26 (1982), 1453

    1982

  4. [10]

    Aguilar & J

    A.C. Aguilar & J. Papavassiliou, J. High Energy Phys. 12 (2006), 012

    2006

  5. [11]

    Fischer, A

    C.S. Fischer, A. Maas & J.M. Pawlowski, Annals Phys. 324 (2009), 2408

    2009

  6. [12]

    Aguilar, D

    A.C. Aguilar, D. Binosi & J. Papavassiliou, Phys. Rev. D84 (2011), 085026

    2011

  7. [13]

    Binosi, D

    D. Binosi, D. Ibanez & J. Papavassiliou, Phys. Rev. D86 (2012), 085033

    2012

  8. [14]

    Aguilar, D

    A.C. Aguilar, D. Binosi & J. Papavassiliou, Phys. Rev. D89 (2014), 085032

    2014

  9. [15]

    Aguilar, M.N

    A.C. Aguilar, M.N. Ferreira, C.T. Figueiredo & J. Papavassiliou, Phys. Rev. D100 (2019), 094039

    2019

  10. [16]

    Curci & R

    G. Curci & R. Ferrari, Nuovo Cim. A32 (1976), 151

    1976

  11. [17]

    Pel´ aez, M

    M. Pel´ aez, M. Tissier & N. Wschebor, Phys. Rev. D88 (2013), 125003

    2013

  12. [18]

    Pel´ aez, M

    M. Pel´ aez, M. Tissier & N. Wschebor, Phys. Rev. D90 (2014), 065031

    2014

  13. [19]

    Reinosa, J

    U. Reinosa, J. Serreau, M. Tissier & N. Wschebor, Phys. Rev. D96 (2017), 014005

    2017

  14. [20]

    Gracey, M

    J.A. Gracey, M. Pel´ aez, U. Reinosa & M. Tissier, Phys. Rev. D100 (2019), 034023

    2019

  15. [21]

    Serreau & M

    J. Serreau & M. Tissier, Phys. Lett. B712 (2012), 97

    2012

  16. [22]

    Serreau, M

    J. Serreau, M. Tissier & A. Tresmontant, Phys. Rev. D89 (2014), 125019

    2014

  17. [23]

    Serreau, M

    J. Serreau, M. Tissier & A. Tresmontant, Phys. Rev. D92 (2015), 105003

    2015

  18. [24]

    Tissier, Phys

    M. Tissier, Phys. Lett. B784 (2018), 146

    2018

  19. [25]

    Gribov, Nucl

    V.N. Gribov, Nucl. Phys. B139 (1978), 1

    1978

  20. [26]

    Zwanziger, Nucl

    D. Zwanziger, Nucl. Phys. B345 (1990), 461

    1990

  21. [27]

    Lavelle, D

    M. Lavelle, D. McMullan & P. Sharma, Phys. Rev. D85 (2012), 045013

    2012

  22. [28]

    Delbourgo & P.D

    R. Delbourgo & P.D. Jarvis, J. Phys. A15 (1982), 611

    1982

  23. [29]

    Wschebor, Int

    N. Wschebor, Int. J. Mod. Phys. A23 (2008), 2961

    2008

  24. [30]

    Tissier & N

    M. Tissier & N. Wschebor, Phys. Rev. D79 (2009), 065008. 32

    2009

  25. [32]

    Browne, D

    R.E. Browne, D. Dudal, J.A. Gracey, V.E.R. Lemes, M.S. Sarandy, R.F. Sobreiro, S.P. Sorella & H. Verschelde, J. Phys. A39 (2006), 7889

    2006

  26. [33]

    Doria, F.A.B

    R.M. Doria, F.A.B. Rabelo de Carvalho & S.P. Sorella, Braz. J. Phys. 20 (1990), 316

    1990

  27. [34]

    Dudal, H

    D. Dudal, H. Verschelde & S.P. Sorella, Phys. Lett. B555 (2003), 126

    2003

  28. [35]

    Maggiore & M

    N. Maggiore & M. Schaden, Phys. Rev. D50 (1994), 6616

    1994

  29. [36]

    Dudal, R.F

    D. Dudal, R.F. Sobreiro, S.P. Sorella & H. Verschelde, Phys. Rev. D72 (2005), 014016

    2005

  30. [37]

    However that would involve the transverse projector and hence necessitate tedious integration by parts for the dipole component

    that the ghost-gluon vertex is finite in the Landau gauge which would only require the ghost 16 and gluon field renormalizations. However that would involve the transverse projector and hence necessitate tedious integration by parts for the dipole component. While the powerful...

    2015

  31. [38]

    Taylor, Nucl

    J.C. Taylor, Nucl. Phys. B33 (1971), 436

    1971

  32. [39]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin & J.H. K¨ uhn, Phys. Rev. Lett. 118 (2017), 082002

    2017

  33. [40]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren & A. Vogt, J. High Energy Phys. 02 (2017), 090

    2017

  34. [41]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard & Y. Schr¨ oder, J. High Energy Phys. 10 (2017), 166

    2017

  35. [42]

    Chetyrkin, G

    K.G. Chetyrkin, G. Falcioni, F. Herzog & J.A.M. Vermaseren, J. High Energy Phys.10 (2017), 179

    2017

  36. [43]

    van Ritbergen, J.A.M

    T. van Ritbergen, J.A.M. Vermaseren & S.A. Larin, Phys. Lett. B400 (1997), 379

    1997

  37. [44]

    Czakon, Nucl

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

    2005

  38. [45]

    T. Ueda, B. Ruijl & J.A.M. Vermaseren, PoS LL2016 (2016), 070

    2016

  39. [46]

    T. Ueda, B. Ruijl & J.A.M. Vermaseren, Comput. Phys. Commun. 253 (2020), 107198

    2020

  40. [47]

    Vermaseren, math-ph/0010025

    J.A.M. Vermaseren, math-ph/0010025

    arXiv

  41. [48]

    Tentyukov & J.A.M

    M. Tentyukov & J.A.M. Vermaseren, Comput. Phys. Commun. 181 (2010), 1419

    2010

  42. [49]

    von Smekal, K

    L. von Smekal, K. Maltman & A. Sternbeck, Phys. Lett. B681 (2009), 336

    2009

  43. [50]

    Slavnov, Theor

    A.A. Slavnov, Theor. Math. Phys. 10 (1972), 153

    1972

  44. [51]

    ’t Hooft, Nucl

    G. ’t Hooft, Nucl. Phys. B61 (1973), 455

    1973

  45. [52]

    Gracey, R.H

    J.A. Gracey, R.H. Mason, T.A. Ryttov & R.M. Simms, Phys. Rev. D108 (2023), 045006

    2023

  46. [53]

    Curci & R

    G. Curci & R. Ferrari, Phys. Lett. B63 (1976), 91

    1976

  47. [54]

    Ojima, Prog

    I. Ojima, Prog. Theor. Phys. 64 (1980), 625

    1980

  48. [55]

    Baulieu & J

    L. Baulieu & J. Thierry-Mieg, Nucl. Phys. B197 (1982), 477

    1982

  49. [56]

    Delduc & S.P

    F. Delduc & S.P. Sorella, Phys. Lett. B231 (1989), 408

    1989

  50. [57]

    Larin & J.A.M

    S.A. Larin & J.A.M. Vermaseren, Phys. Lett. B303 (1993), 334

    1993

  51. [59]

    Ruijl, T

    B. Ruijl, T. Ueda, J.A.M. Vermaseren & A. Vogt, J. High Energy Phys. 06 (2017), 040

    2017

  52. [60]

    Gracey, J

    J.A. Gracey, J. Phys. A46 (2013), 225403; J. Phys. A48 (2015), 119501(E). 33

    2013

  53. [61]

    Gracey & R.H

    J.A. Gracey & R.H. Mason, J. Phys. A56 (2023), 085401

    2023

  54. [62]

    Nachtmann & W

    O. Nachtmann & W. Wetzel, Nucl. Phys. B187 (1981), 333

    1981

  55. [63]

    Tarrach, Nucl

    R. Tarrach, Nucl. Phys. B183 (1981), 384

    1981

  56. [64]

    Tarasov, Phys

    O.V. Tarasov, Phys. Part. Nucl. Lett. 17 (2020), 109

    2020

  57. [65]

    Chetyrkin, Phys

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

    1997

  58. [66]

    Vermaseren, S.A

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

    1997

  59. [67]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin & J.H. K¨ uhn, J. High Energy Phys. 10 (2014), 076

    2014

  60. [68]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard & Y. Schr¨ oder, J. High Energy Phys. 01 (2017), 081

    2017

  61. [69]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin & J.H. K¨ uhn, J. High Energy Phys. 04 (2017), 119

    2017

  62. [70]

    Baikov and K.G

    P.A. Baikov and K.G. Chetyrkin, J. High Energy Phys. 06 (2018), 141

    2018

  63. [71]

    Baikov & K.G

    P.A. Baikov & K.G. Chetyrkin, J. High Energy Phys. 10 (2019), 190

    2019

  64. [72]

    Jack, Phys

    I. Jack, Phys. Rev. D109 (2024), 045007

    2024

  65. [73]

    Braaten & J.P

    E. Braaten & J.P. Leveille, Phys. Rev. D24 (1981), 1369

    1981

  66. [74]

    Gracey, Phys

    J.A. Gracey, Phys. Rev. D110 (2024), 045015

    2024

  67. [75]

    Laporta, Phys

    S. Laporta, Phys. Lett. B549 (2002), 115

    2002

  68. [76]

    Gross & F.J

    D.J. Gross & F.J. Wilczek, Phys. Rev. Lett. 30 (1973), 1343

    1973

  69. [77]

    Politzer, Phys

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

    1973

  70. [78]

    Caswell, Phys

    W.E. Caswell, Phys. Rev. Lett. 33 (1974), 244

    1974

  71. [79]

    Jones, Nucl

    D.R.T. Jones, Nucl. Phys. B75 (1974), 531

    1974

  72. [80]

    Tarasov, A.A

    O.V. Tarasov, A.A. Vladimirov & A.Yu. Zharkov, Phys. Lett. B93 (1980), 429

    1980

  73. [81]

    Vladimirov, Theor

    A.A. Vladimirov, Theor. Math. Phys. 43 (1980), 417

    1980

  74. [82]

    Chetyrkin, A.L

    K.G. Chetyrkin, A.L. Kataev & F.V. Tkachov, Nucl. Phys. B174 (1980), 345

    1980

  75. [83]

    Herzog, Nucl

    F. Herzog, Nucl. Phys. B926 (2018), 370

    2018

  76. [84]

    Chetyrkin, G

    K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren, PoS RADCOR2017 (2018), 004

    2018

  77. [85]

    Ferguson, D.H

    H.R.P. Ferguson, D.H. Bailey & S. Arno, Math. Comput. 68 (1999), 351

    1999

  78. [86]

    Kompaniets & E

    M.V. Kompaniets & E. Panzer, Phys. Rev. D96 (2017), 036016

    2017

  79. [87]

    Schnetz, Phys

    O. Schnetz, Phys. Rev. D97 (2018), 085018

    2018

  80. [88]

    Schnetz, Commun

    O. Schnetz, Commun. Num. Theor. Phys. 08 (2014), 589

    2014

  81. [89]

    Schnetz & S

    O. Schnetz & S. Theil, PoS LL2024 (2024), 026

    2024

  82. [90]

    Luthe, A

    T. Luthe, A. Maier, P. Marquard & Y. Schr¨ oder, J. High Energy Phys. 03 (2017), 020. 34

    2017

  83. [91]

    Misiak & M

    M. Misiak & M. M¨ unz, Phys. Lett. B344 (1995), 308

    1995

  84. [92]

    Chetyrkin, M

    K.G. Chetyrkin, M. Misiak & M. M¨ unz, Nucl. Phys. B518 (1998), 473

    1998

  85. [93]

    Herzog, S

    F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren & A. Vogt, Phys. Lett. B790 (2019), 436

    2019

  86. [94]

    van Ritbergen, A.N

    T. van Ritbergen, A.N. Schellekens & J.A.M. Vermaseren, Int. J. Mod. Phys. A14 (1999), 41. 35

    1999

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