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arxiv: 2604.09534 · v2 · submitted 2026-04-10 · ✦ hep-ph · hep-th

Recognition: 2 theorem links

· Lean Theorem

The four-loop non-singlet splitting functions in QCD

Thomas Gehrmann , Andreas von Manteuffel , Vasily Sotnikov , Tong-Zhi Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:39 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords QCDsplitting functionsnon-singletfour-loopparton distributionsanomalous dimensionsDGLAP evolutionperturbative QCD
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0 comments X

The pith

Analytic expressions for all four-loop non-singlet splitting functions in QCD are now available.

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

The paper computes the four-loop splitting functions that control how non-singlet quark distributions evolve with energy scale in perturbative QCD. These are the first fully analytic results for every non-singlet contribution at this order. The expressions also deliver analytic forms of the associated four-loop virtual and rapidity anomalous dimensions needed for resummation. Numerical versions are supplied so the functions can be used directly in evolution codes. A reader would care because accurate higher-order splitting improves predictions for parton distributions in high-energy collisions.

Core claim

We obtain, for the first time, fully analytic expressions for all non-singlet contributions at four-loop order. These confirm previous partial results and allow extraction of the analytic four-loop virtual and rapidity anomalous dimensions that enter logarithmic resummation. Precise numerical representations suitable for parton evolution are also provided.

What carries the argument

The four-loop non-singlet splitting functions in QCD, obtained via diagram reduction and integration methods.

If this is right

  • Non-singlet parton distributions can now be evolved to four-loop accuracy in scale.
  • Analytic four-loop virtual and rapidity anomalous dimensions become available for resummation calculations.
  • Numerical implementations of these splitting functions can be inserted into existing parton-evolution programs.

Where Pith is reading between the lines

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

  • Global fits of parton distributions could incorporate these higher-order effects to reduce theoretical uncertainty.
  • Future precision measurements in deep-inelastic scattering might directly confront the four-loop predictions.
  • The same computational approach could be applied to the still-missing four-loop singlet splitting functions.

Load-bearing premise

The perturbative expansion and the standard methods for reducing diagrams and evaluating integrals remain valid and controllable at four loops without unforeseen cancellations or divergences.

What would settle it

An independent numerical evaluation of any four-loop non-singlet splitting function that differs from these analytic expressions, or a mismatch with data on non-singlet evolution at very high scales, would settle the claim.

Figures

Figures reproduced from arXiv: 2604.09534 by Andreas von Manteuffel, Thomas Gehrmann, Tong-Zhi Yang, Vasily Sotnikov.

Figure 1
Figure 1. Figure 1: FIG. 1. Sample Feynman diagrams for off-shell quark self energies with an operator insertion (crossed vertex) contributing to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Feynman integral topology with 11 denominators [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The four-loop non-singlet splitting functions [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The scale evolution of parton distributions is governed by splitting functions. We compute the four-loop splitting functions in perturbative QCD that control the evolution of quark non-singlet distributions. We confirm previous partial results and obtain, for the first time, fully analytic expressions for all non-singlet contributions at this order. These allow us to extract the analytic form of the four-loop virtual and rapidity anomalous dimensions entering logarithmic resummation. We provide precise numerical representations of the splitting functions suitable for parton evolution.

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

0 major / 2 minor

Summary. The paper reports the calculation of the four-loop non-singlet splitting functions in QCD. It confirms all previously available partial results and provides the first complete analytic expressions for the non-singlet splitting functions at four-loop order. Additionally, the four-loop virtual and rapidity anomalous dimensions are extracted, and numerical representations of the splitting functions are given for practical use in parton distribution evolution.

Significance. This computation is of high significance for the QCD community. Completing the four-loop non-singlet splitting functions allows for more accurate evolution of parton distributions in global fits and for higher-order predictions in collider physics. The analytic nature of the results is particularly useful for understanding the structure of higher-order corrections and for applications in resummation. The confirmation of prior partial results provides an important consistency check.

minor comments (2)
  1. It would be helpful to specify the references for the 'previous partial results' that are confirmed in the abstract.
  2. The notation for the splitting functions and anomalous dimensions could be introduced more explicitly in the introductory sections to aid readers not familiar with the conventions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the four-loop non-singlet splitting functions in QCD and for recommending minor revision. We appreciate the recognition of the work's significance for the QCD community, including its confirmation of prior partial results and provision of the first complete analytic expressions. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation is a direct perturbative computation of four-loop non-singlet splitting functions starting from Feynman diagrams, followed by diagram reduction, integration-by-parts identities, and master-integral evaluations to obtain analytic expressions. Prior partial results are confirmed as an internal consistency check rather than serving as load-bearing inputs, and the new fully analytic forms are extracted independently. No parameters are fitted in a manner that renders subsequent predictions tautological, no self-definitional relations equate outputs to inputs by construction, and self-citations (if present) support methodological details without reducing the central four-loop claim to an unverified prior assertion. The computation remains self-contained against external benchmarks at lower orders.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

pith-pipeline@v0.9.0 · 5379 in / 906 out tokens · 39715 ms · 2026-05-10T16:39:48.761806+00:00 · methodology

discussion (0)

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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.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part

    hep-ph 2026-05 unverdicted novelty 7.0

    Closed-form expression for the ζ(3) term of the four-loop non-singlet twist-two quark anomalous dimension for arbitrary N, extracted via analytic reconstruction from Mellin moments.

  2. Properties and implications of the four-loop non-singlet splitting functions in QCD

    hep-ph 2026-05 unverdicted novelty 7.0

    Four-loop non-singlet QCD splitting functions are verified for consistency and used to finalize analytical forms for the gluon virtual anomalous dimension and N^4LL threshold resummation coefficients, revealing a new ...

  3. Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

    hep-ph 2026-04 unverdicted novelty 7.0

    The zeta(3)-proportional contribution to the four-loop twist-two gluon anomalous dimension gamma_gg^(3)(N) is constructed in analytic form from low-N moments via the LLL algorithm together with reciprocity and supersy...

Reference graph

Works this paper leans on

99 extracted references · 81 canonical work pages · cited by 3 Pith papers

  1. [1]

    J. D. Bjorken, Asymptotic Sum Rules at Infinite Momen- tum, Phys. Rev.179, 1547 (1969)

    1969

  2. [2]

    J. D. Bjorken and E. A. Paschos, Inelastic Electron Pro- ton and gamma Proton Scattering, and the Structure of the Nucleon, Phys. Rev.185, 1975 (1969)

    1975

  3. [3]

    G.AltarelliandG.Parisi,AsymptoticFreedominParton Language, Nucl. Phys. B126, 298 (1977)

    1977

  4. [4]

    V. N. Gribov and L. N. Lipatov, Deep inelastic ep scat- tering in perturbation theory, Sov. J. Nucl. Phys.15, 438 (1972)

    1972

  5. [5]

    Y. L. Dokshitzer, Calculation of the Structure Func- tionsforDeepInelasticScatteringande +e− Annihilation by Perturbation Theory in Quantum Chromodynamics., Sov. Phys. JETP46, 641 (1977)

    1977

  6. [6]

    D. J. Gross and F. Wilczek, Asymptotically free gauge theories. 2., Phys. Rev. D9, 980 (1974)

    1974

  7. [7]

    H. D. Politzer, Asymptotic Freedom: An Approach to Strong Interactions, Phys. Rept.14, 129 (1974)

    1974

  8. [8]

    E. G. Floratos, D. A. Ross, and C. T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories: The Anomalous Dimensions of Wilson Operators, Nucl. Phys. B129, 66 (1977), [Erratum: Nucl.Phys.B 139, 545–546 (1978)]

    1977

  9. [9]

    E. G. Floratos, D. A. Ross, and C. T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories. 2. Flavor Singlet Wilson Operators and Coefficient Func- tions, Nucl. Phys. B152, 493 (1979)

    1979

  10. [10]

    Gonzalez-Arroyo, C

    A. Gonzalez-Arroyo, C. Lopez, and F. J. Yndurain, Sec- ond Order Contributions to the Structure Functions in Deep Inelastic Scattering. 1. Theoretical Calculations, Nucl. Phys. B153, 161 (1979)

    1979

  11. [11]

    Curci, W

    G. Curci, W. Furmanski, and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl. Phys. B175, 27 (1980)

    1980

  12. [12]

    Furmanski and R

    W. Furmanski and R. Petronzio, Singlet Parton Densities Beyond Leading Order, Phys. Lett. B97, 437 (1980)

    1980

  13. [13]

    Hamberg and W

    R. Hamberg and W. L. van Neerven, The Correct renor- malization of the gluon operator in a covariant gauge, Nucl. Phys. B379, 143 (1992)

    1992

  14. [14]

    S. Moch, J. A. M. Vermaseren, and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys. B688, 101 (2004), arXiv:hep-ph/0403192

    work page arXiv 2004

  15. [15]

    A. Vogt, S. Moch, and J. A. M. Vermaseren, The Three- loop splitting functions in QCD: The Singlet case, Nucl. Phys. B691, 129 (2004), arXiv:hep-ph/0404111

    work page arXiv 2004

  16. [16]

    Bl¨ umlein, P

    J. Blümlein, P. Marquard, C. Schneider, and K. Schön- wald, The three-loop unpolarized and polarized non- singlet anomalous dimensions from off shell operator matrix elements, Nucl. Phys. B971, 115542 (2021), arXiv:2107.06267 [hep-ph]

    work page arXiv 2021

  17. [17]

    Blümlein, P

    J. Blümlein, P. Marquard, C. Schneider, and K. Schön- wald, The three-loop polarized singlet anomalous dimen- sions from off-shell operator matrix elements, JHEP01, 193, arXiv:2111.12401 [hep-ph]

    work page arXiv

  18. [18]

    Gehrmann, A

    T. Gehrmann, A. vonManteuffel,andT.-Z.Yang,Renor- malization of twist-two operators in covariant gauge to three loops in QCD, JHEP04, 041, arXiv:2302.00022 [hep-ph]

    work page arXiv

  19. [19]

    S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, Four-Loop Non-Singlet Splitting Func- tions in the Planar Limit and Beyond, JHEP10, 041, arXiv:1707.08315 [hep-ph]

    work page arXiv

  20. [20]

    S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, Low moments of the four-loop splitting functions in QCD, Phys. Lett. B825, 136853 (2022), arXiv:2111.15561 [hep-ph]

    work page arXiv 2022

  21. [21]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, and A. Vogt, Four-loop splittingfunctionsinQCD–Thequark-quarkcase,Phys. Lett. B842, 137944 (2023), arXiv:2302.07593 [hep-ph]

    work page arXiv 2023

  22. [22]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, and A. Vogt, Four-loop splitting functions in QCD – The gluon-to-quark case, Phys.Lett.B846,138215(2023),arXiv:2307.04158[hep- ph]

    work page arXiv 2023

  23. [23]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Four-loop splitting functions in QCD – The quark-to-gluon case, Phys. Lett. B856, 138906 (2024), arXiv:2404.09701 [hep-ph]

    work page arXiv 2024

  24. [24]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Four-loop splitting functions in QCD – the gluon-gluon case –, Phys. Lett. B860, 139194 (2025), arXiv:2410.08089 [hep-ph]

    work page arXiv 2025

  25. [25]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Additional results on the four-loop flavour-singlet split- ting functions in QCD, Phys. Lett. B875, 140278 (2026), arXiv:2512.10783 [hep-ph]

    work page arXiv 2026

  26. [26]

    J. A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O (1 / N(f)) in deep inelastic scattering, Phys. Lett. B322, 141 (1994), arXiv:hep-ph/9401214

    work page arXiv 1994

  27. [27]

    Davies, A

    J. Davies, A. Vogt, B. Ruijl, T. Ueda, and J. A. M. Ver- maseren, Large-nf contributions to the four-loop split- ting functions in QCD, Nucl. Phys. B915, 335 (2017), arXiv:1610.07477 [hep-ph]

    work page arXiv 2017

  28. [28]

    Gehrmann, A

    T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T.-Z. Yang, CompleteN 2 f contributions to four- loop pure-singlet splitting functions, JHEP01, 029, arXiv:2308.07958 [hep-ph]

    work page arXiv

  29. [29]

    Gehrmann, A

    T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T.-Z. Yang,TheN f C 3 F contributiontothenon-singletsplitting function at four-loop order, Phys. Lett. B849, 138427 (2024), arXiv:2310.12240 [hep-ph]

    work page arXiv 2024

  30. [30]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, J. Vermaseren, and A. Vogt, The double fermionic contribution to the four- loopquark-to-gluonsplittingfunction,Phys.Lett.B848, 138351 (2024), arXiv:2310.01245 [hep-ph]

    work page arXiv 2024

  31. [31]

    B. A. Kniehl, S. Moch, V. N. Velizhanin, and A. Vogt, Flavor Nonsinglet Splitting Functions at Four Loops in QCD: Fermionic Contributions, Phys. Rev. Lett.135, 071902 (2025), arXiv:2505.09381 [hep-ph]

    work page arXiv 2025

  32. [32]

    B.A.KniehlandV.N.Velizhanin,Four-LoopAnomalous 7 Dimension of Flavor Nonsinglet Twist-Two Operator of General Lorentz Spin in QCD:ζ(3) Term, Phys. Rev. Lett.134, 131901 (2025), arXiv:2503.20422 [hep-ph]

    work page arXiv 2025

  33. [33]

    McGowan, T

    J. McGowan, T. Cridge, L. A. Harland-Lang, and R. S. Thorne, Approximate N3LO parton distribution func- tions with theoretical uncertainties: MSHT20aN 3LO PDFs, Eur. Phys. J. C83, 185 (2023), [Erratum: Eur.Phys.J.C 83, 302 (2023)], arXiv:2207.04739 [hep-ph]

    work page arXiv 2023

  34. [34]

    Cooper-Sarkar, T

    A. Cooper-Sarkar, T. Cridge, F. Giuli, L. A. Harland- Lang, F. Hekhorn, J. Huston, G. Magni, S. Moch, and R. S. Thorne, A Benchmarking of QCD Evolution at Ap- proximateN 3LO, (2024), arXiv:2406.16188 [hep-ph]

    work page arXiv 2024

  35. [35]

    R. D. Ballet al.(NNPDF), The path to N 3LO par- ton distributions, Eur. Phys. J. C84, 659 (2024), arXiv:2402.18635 [hep-ph]

    work page arXiv 2024

  36. [36]

    Cridgeet al., Combination of aN3LO PDFs and impli- cations for Higgs production cross-sections at the LHC, J

    T. Cridgeet al., Combination of aN3LO PDFs and impli- cations for Higgs production cross-sections at the LHC, J. Phys. G52, 6 (2025), arXiv:2411.05373 [hep-ph]

    work page arXiv 2025

  37. [37]

    Hampson and M

    C. Hampson and M. Guzzi, DGLAP evolution at N3LO with the candia algorithm, Phys. Rev. D113, 074008 (2026), arXiv:2512.22667 [hep-ph]

    work page arXiv 2026

  38. [38]

    Karlberg, P

    A. Karlberg, P. Nason, G. Salam, G. Zanderighi, and F. Dreyer, HOPPET v2.0.0 release note, Eur. Phys. J. C 86, 157 (2026), arXiv:2510.09310 [hep-ph]

    work page arXiv 2026

  39. [39]

    Y. L. Dokshitzer, G. Marchesini, and G. P. Salam, Revis- iting parton evolution and the large-x limit, Phys. Lett. B634, 504 (2006), arXiv:hep-ph/0511302

    work page arXiv 2006

  40. [40]

    Mitov, S

    A. Mitov, S. Moch, and A. Vogt, Next-to-Next-to- Leading Order Evolution of Non-Singlet Fragmentation Functions, Phys. Lett. B638, 61 (2006), arXiv:hep- ph/0604053

    work page arXiv 2006

  41. [41]

    Basso and G.P

    B. Basso and G. P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys. B775, 1 (2007), arXiv:hep-th/0612247

    work page arXiv 2007

  42. [42]

    J. A. Dixon and J. C. Taylor, Renormalization of Wilson operatorsingaugetheories,Nucl.Phys.B78,552(1974)

    1974

  43. [43]

    Falcioni and F

    G. Falcioni and F. Herzog, Renormalization of gluonic leading-twist operators in covariant gauges, JHEP05, 177, arXiv:2203.11181 [hep-ph]

    work page arXiv

  44. [44]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, and S. Van Thurenhout, Constraints for twist-two alien operators in QCD, JHEP 11, 080, arXiv:2409.02870 [hep-ph]

    work page arXiv

  45. [45]

    Gehrmann, A

    T. Gehrmann, A. von Manteuffel, and T.-Z. Yang, Leading Twist-Two Gauge-Variant Counterterms, PoS LL2024, 087 (2024), arXiv:2409.10303 [hep-ph]

    work page arXiv 2024

  46. [46]

    Nogueira, Automatic Feynman graph generation, J

    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys.105, 279 (1993)

    1993

  47. [47]

    J. A. M. Vermaseren, New features of FORM, (2000), arXiv:math-ph/0010025

    work page arXiv 2000

  48. [48]

    Davies, T

    J. Davies, T. Kaneko, C. Marinissen, T. Ueda, and J. A. M. Vermaseren, FORM Version 5.0, (2026), arXiv:2601.19982 [hep-ph]

    work page arXiv 2026

  49. [49]

    van Ritbergen, A

    T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren, Group theory factors for Feynman dia- grams, Int. J. Mod. Phys. A14, 41 (1999), arXiv:hep- ph/9802376

    work page arXiv 1999

  50. [50]

    Ablinger, J

    J. Ablinger, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, and F. Wissbrock, Massive 3-loop Lad- der Diagrams for Quarkonic Local Operator Matrix El- ements, Nucl. Phys. B864, 52 (2012), arXiv:1206.2252 [hep-ph]

    work page arXiv 2012

  51. [51]

    Ablinger, A

    J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, and C. Schneider, The 3-loop pure singlet heavy flavor contributions to the structure func- tionF 2(x, Q2)and the anomalous dimension, Nucl. Phys. B890, 48 (2014), arXiv:1409.1135 [hep-ph]

    work page arXiv 2014

  52. [52]

    K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique, Nucl. Phys. B174, 345 (1980)

    1980

  53. [53]

    High-precision calculation of multi-loop Feynman integrals by difference equations

    S. Laporta, High precision calculation of multiloop Feyn- man integrals by difference equations, Int. J. Mod. Phys. A15, 5087 (2000), arXiv:hep-ph/0102033

    work page Pith review arXiv 2000

  54. [54]

    Reduze 2 - Distributed Feynman Integral Reduction

    A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, (2012), arXiv:1201.4330 [hep-ph]

    work page Pith review arXiv 2012

  55. [55]

    A novel approach to integration by parts reduction

    A. von Manteuffel and R. M. Schabinger, A novel ap- proach to integration by parts reduction, Phys. Lett. B 744, 101 (2015), arXiv:1406.4513 [hep-ph]

    work page Pith review arXiv 2015

  56. [56]

    Scattering amplitudes over finite fields and multivariate functional reconstruction

    T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP12, 030, arXiv:1608.01902 [hep-ph]

    work page Pith review arXiv

  57. [57]

    Driesse, G.U

    M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self- Force Order, Phys. Rev. Lett.132, 241402 (2024), arXiv:2403.07781 [hep-th]

    work page arXiv 2024

  58. [58]

    X. Guan, X. Liu, Y.-Q. Ma, and W.-H. Wu, Blade: A package for block-triangular form improved Feynman in- tegrals decomposition, Comput. Phys. Commun.310, 109538 (2025), arXiv:2405.14621 [hep-ph]

    work page arXiv 2025

  59. [59]

    Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov, and M. Zeng, Amplitudes, su- persymmetric black hole scattering atO G5 , and loop integration, JHEP10, 023, arXiv:2406.01554 [hep-th]

    work page arXiv

  60. [60]

    von Hippel and M

    M. von Hippel and M. Wilhelm, Refining Integration- by-Parts Reduction of Feynman Integrals with Machine Learning, JHEP05, 185, arXiv:2502.05121 [hep-th]

    work page arXiv

  61. [61]

    Song, T.-Z

    Z.-Y. Song, T.-Z. Yang, Q.-H. Cao, M.-x. Luo, and H. X. Zhu, Explainable AI-assisted Optimization for Feynman Integral Reduction, (2025), arXiv:2502.09544 [hep-ph]

    work page arXiv 2025

  62. [62]

    Lange, J

    F. Lange, J. Usovitsch, and Z. Wu, Kira 3: integral re- duction with efficient seeding and optimized equation se- lection, Comput. Phys. Commun.322, 109999 (2026), arXiv:2505.20197 [hep-ph]

    work page arXiv 2026

  63. [63]

    K. J. Larsen and Y. Zhang, Integration-by-parts reduc- tions from unitarity cuts and algebraic geometry, Phys. Rev. D93, 041701 (2016), arXiv:1511.01071 [hep-th]

    work page arXiv 2016

  64. [64]

    A. V. Smirnov and V. A. Smirnov, How to choose master integrals, Nucl. Phys. B960, 115213 (2020), arXiv:2002.08042 [hep-ph]

    work page arXiv 2020

  65. [65]

    Usovitsch,Factorization of denominators in integration-by-parts reductions, 2002.08173

    J. Usovitsch, Factorization of denominators in integration-by-parts reductions, (2020), arXiv:2002.08173 [hep-ph]

    work page arXiv 2020

  66. [66]

    Abreu, J

    S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, and B. Page, Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD, Phys. Rev. Lett.122, 082002 (2019), arXiv:1812.04586 [hep-ph]

    work page arXiv 2019

  67. [67]

    Heller and A

    M. Heller and A. von Manteuffel, MultivariateApart: Generalized partial fractions, Comput. Phys. Commun. 271, 108174 (2022), arXiv:2101.08283 [cs.SC]

    work page arXiv 2022

  68. [68]

    Gehrmann and E

    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580, 485 (2000), arXiv:hep-ph/9912329

    work page arXiv 2000

  69. [69]

    J. M. Henn, Multiloop integrals in dimensional regu- larization made simple, Phys. Rev. Lett.110, 251601 (2013), arXiv:1304.1806 [hep-th]

    work page arXiv 2013

  70. [70]

    Blümlein, M

    J. Blümlein, M. Kauers, S. Klein, and C. Schneider, De- 8 termining the closed forms of the O(α3 s) anomalous di- mensions and Wilson coefficients from Mellin moments by means of computer algebra, Comput. Phys. Commun. 180, 2143 (2009), arXiv:0902.4091 [hep-ph]

    work page arXiv 2009

  71. [71]

    Ablinger, A

    J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, and C. Schneider, Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra, Comput. Phys. Commun.202, 33 (2016), arXiv:1509.08324 [hep-ph]

    work page arXiv 2016

  72. [72]

    R.N.Lee, A.V.Smirnov,andV.A.Smirnov,Solvingdif- ferential equations for Feynman integrals by expansions near singular points, JHEP03, 008, arXiv:1709.07525 [hep-ph]

    work page arXiv

  73. [73]

    P. A. Baikov and K. G. Chetyrkin, Four Loop Massless Propagators: An Algebraic Evaluation of All Master In- tegrals, Nucl. Phys. B837, 186 (2010), arXiv:1004.1153 [hep-ph]

    work page arXiv 2010

  74. [74]

    R. N. Lee, A. V. Smirnov, and V. A. Smirnov, Mas- ter Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys. B856, 95 (2012), arXiv:1108.0732 [hep-th]

    work page arXiv 2012

  75. [75]

    von Manteuffel and R

    A. von Manteuffel and R. M. Schabinger, Planar mas- ter integrals for four-loop form factors, JHEP05, 073, arXiv:1903.06171 [hep-ph]

    work page arXiv 1903

  76. [76]

    R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Master in- tegrals for four-loop massless form factors, Eur. Phys. J. C83, 1041 (2023), arXiv:2309.00054 [hep-ph]

    work page arXiv 2023

  77. [77]

    J. A. M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A14, 2037 (1999), arXiv:hep-ph/9806280

    work page Pith review arXiv 2037

  78. [78]

    J. Henn, B. Mistlberger, V. A. Smirnov, and P. Wasser, Constructing d-log integrands and computing master in- tegrals for three-loop four-particle scattering, JHEP04, 167, arXiv:2002.09492 [hep-ph]

    work page arXiv 2002

  79. [79]

    Bogner, S

    C. Bogner, S. Müller-Stach, and S. Weinzierl, The un- equal mass sunrise integral expressed through iterated integrals on M1,3, Nucl. Phys. B954, 114991 (2020), arXiv:1907.01251 [hep-th]

    work page arXiv 2020

  80. [80]

    A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

    J. Ablinger,A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics, Master’s thesis, Linz U. (2009), arXiv:1011.1176 [math-ph]

    work page Pith review arXiv 2009

Showing first 80 references.