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REVIEW 4 major objections 4 minor 50 references

A direct four-loop computation in the unbroken Standard Model reproduces the gauge-coupling beta functions obtained from Weyl consistency conditions and supplies the first four-loop gauge-field anomalous dimensions.

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2026-08-01 06:58 UTC pith:KKOZX3SX

load-bearing objection Serious four-loop calculation; the new gauge-field anomalous dimensions are the real result, but the beta-function 'independent confirmation' is partly achieved by calibrating the gamma5 prescription to the known answer. the 4 major comments →

arxiv 2607.21586 v1 pith:KKOZX3SX submitted 2026-07-23 hep-ph

Gauge coupling beta functions and gauge field anomalous dimensions at four loops in the Standard Model

M.A.Bezuglov , B.A.Kniehl , V.N.Velizhanin This is my paper
classification hep-ph PACS 11.10.Hi12.15.-y12.38.Bx
keywords four-loop beta functionsStandard Modelgauge couplingsgauge-field anomalous dimensionsrenormalization groupgamma5 schemebackground field methodWeyl consistency conditions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the four-loop renormalization of the gauge sector of the unbroken Standard Model—masses omitted—can be obtained by a direct, diagram-by-diagram Feynman calculation, rather than only by consistency arguments. Using the background-field gauge and a specific γ5 reading-point prescription, the authors compute the four-loop gauge-field anomalous dimensions and from them the gauge-coupling beta functions. They report full agreement between their beta functions and those previously derived from Weyl consistency conditions, and they present the four-loop gauge-field anomalous dimensions as new results. This matters because running couplings at very high energies—used in vacuum-stability, unification, and threshold studies—need four-loop accuracy, and a direct computation is the strongest check that the earlier consistency-based results are physically correct.

Core claim

The central claim, stated on the paper's own terms, is that the four-loop gauge coupling beta functions of the Standard Model in the unbroken phase are those given in Eqs. (30)–(32), fully agreeing with the WCC-based results, and that the four-loop anomalous dimensions of the gauge fields, Eqs. (33)–(34) together with γ_BB = −β_1, are new. The calculation is performed directly from the Standard Model Lagrangian in background-field gauge, keeping all gauge parameters arbitrary, and the γ5-dependent pieces are fixed by reading γ5 at internal vertices and averaging over the 3×3 internal-vertex choices. If correct, this is the first direct diagrammatic derivation of the complete four-loop gauge

What carries the argument

The carrying mechanism is the background-field Ward identity Z_{g_i} = Z_{\hat V_i}^{-1/2}, which turns gauge-field renormalization constants directly into gauge-coupling renormalization constants, so the entire four-loop problem reduces to two-point self-energy diagrams. The chiral-sector machinery is the reading-point prescription for γ5: within each fermion trace containing an odd number of γ5 insertions, γ5 is read at an internal vertex, and the results are averaged over the 3×3 internal-vertex choices; this averaging exactly recovers the known excess terms in Eqs. (26)–(28). Automated generation of four-loop propagator diagrams and reduction of the resulting integrals complete the pipel

Load-bearing premise

Everything rests on the assumption that the γ5 reading-point prescription—reading γ5 at an internal vertex and averaging over the 3×3 internal-vertex choices—is the unique physical resolution of the γ5 ambiguity in four-loop dimensional regularization; with a different resolution the four-loop beta functions and anomalous dimensions would be scheme dependent.

What would settle it

Recompute the four-loop gauge-field anomalous dimensions with an independent, well-defined treatment of γ5 in dimensional regularization—for example, a different reading-point choice or a different consistent scheme—and check whether the resulting beta functions still satisfy the background-field Ward identity, including γ_BB = −β_1. If those relations fail or the four-loop results change, the claimed scheme independence of the four-loop gauge sector is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The four-loop gauge beta functions of the Standard Model now have a direct, diagram-by-diagram derivation, independently confirming the WCC-based values.
  • The previously unknown four-loop gauge-field anomalous dimensions, including their gauge-parameter dependence, complete the four-loop renormalization of the SM gauge sector in the unbroken phase.
  • Precision renormalization-group studies—running couplings, threshold matching, gauge unification, and electroweak vacuum stability—can incorporate four-loop gauge corrections instead of stopping at three-loop gauge order.
  • The exact recovery of the known excess terms under the 3×3 internal-vertex averaging validates the internal-vertex reading-point choice as a consistent γ5 scheme at four loops, at least in the gauge sector.

Where Pith is reading between the lines

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

  • If the paper is right, the same internal-vertex averaging method should also apply to the still-missing four-loop Yukawa and Higgs beta functions, completing the SM's four-loop renormalization-group functions.
  • If the paper is right, four-loop gauge corrections will shift numerical predictions for gauge-coupling unification contours and the electroweak vacuum-stability boundary by amounts not captured in current three-loop analyses.
  • If the paper is right, the three-vector fermion-triangle cancellation is a special consequence of N_c=3; SM extensions with different color multiplicities or additional fermions would require a fresh four-loop gauge calculation, and γ5 scheme dependence could resurface there.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper presents a four-loop calculation of the Standard Model gauge-coupling beta functions and gauge-field anomalous dimensions in the unbroken phase, using the background-field method and a reading-point prescription for gamma5. The four-loop beta functions in Eqs. (30)–(32) are reported to agree exactly with the WCC-based results of Refs. [32,33], and the four-loop gauge-field anomalous dimensions in Eqs. (33)–(34) are presented as new. The calculation is performed in arbitrary gauge, with detailed diagram counts and ancillary files for the full R_xi results.

Significance. If correct, this is a substantial technical achievement: a direct four-loop diagrammatic calculation of the complete SM gauge sector, including new anomalous dimensions, with a large number of diagrams evaluated in general gauge. The availability of ancillary files with the full renormalization constants is a strength. However, the gamma5-dependent part of the calculation is explicitly calibrated to reproduce the previously known WCC-based beta functions, so the claimed 'independent confirmation' is not established. The new anomalous dimensions inherit the same calibrated scheme and lack an external cross-check, which limits the strength of the paper's central claim.

major comments (4)
  1. [Section 3, Eqs. (26)–(28)] The adoption of the gamma5 reading-point prescription is not an independent check but a fit: the text states that choosing internal vertices and averaging over the 3×3 internal-vertex choices 'exactly recover[s] the excess contributions' in Eqs. (26)–(28). Since the prescription is selected by requiring exact reproduction of the WCC-based differences, the agreement of Eqs. (30)–(32) with Refs. [32,33] is achieved by construction. The Introduction and Conclusion claims of 'independently confirms' are therefore overstated. Please reframe the beta-function agreement as a consistency check in a particular gamma5 scheme, or provide an independent argument for the uniqueness of the 3×3 average.
  2. [Section 4, Eqs. (33)–(34)] The new four-loop gauge-field anomalous dimensions are computed in the same calibrated gamma5 scheme, and no external cross-check is provided. Because Section 3 explicitly reports that different reading-point choices give different results, the anomalous dimensions are scheme-dependent unless the 3×3 internal-vertex average is proven to select the unique physical result. Please add a concrete test — for example, WCC-type constraints on the anomalous dimensions, gauge independence of physical quantities, or a comparison with an alternative gamma5 scheme — before presenting these as the four-loop anomalous dimensions.
  3. [Introduction and Section 3] The Introduction states that 'the final results are independent of the auxiliary choices introduced by this procedure,' but Section 3 states that different reading-point choices give different results and that only the averaged internal-vertex result matches the known excess. These statements are in direct tension. Please specify precisely what is independent (for example, the averaged result) or remove the independence claim.
  4. [Section 3, Fig. 1(c)] The exclusion of fermion-triangle diagrams with three vector lines relies on the assertion that such contributions sum to zero for the SM field content with Nc=3. No cancellation identity or reference is given for this nontrivial statement. Please provide the explicit sum or a reference so the reader can verify that no finite contribution is being omitted.
minor comments (4)
  1. [Section 3, Eqs. (26)–(28)] The symbols y_u, y_d, y_l and their powers are used without definition. Please clarify their relation to the Yukawa matrices Y_u, Y_d, Y_l (e.g., traces of Y†Y) and consistently define the notation used in Eqs. (26)–(28) and in the final results.
  2. [Section 3] The phrase 'gaugeless limit' is introduced without explanation. Please define it precisely, including which couplings are set to zero.
  3. [Table 1] The FORCER topologies 'Mno1' and 'Mhaha' are mentioned but not explained. A one-sentence description of these topology classes would help the reader interpret the diagram counts and the later statement that only these topologies contribute.
  4. [General] There are minor typographical and formatting issues, e.g., 'Fadeev–Popov' should be 'Faddeev–Popov', and the quoted 'na ¨ıve' contains a formatting artifact. Also, the sign convention in Eq. (25) could be stated more explicitly for the beta-function coefficients.

Circularity Check

2 steps flagged

The γ5 reading-point scheme is calibrated to the WCC beta functions, so the 'full agreement' with Refs. [32,33] is partly by construction and the new gauge-field anomalous dimensions inherit that calibrated scheme.

specific steps
  1. fitted input called prediction [Section 3, paragraph following Eqs. (26)-(28); Section 5 Conclusion]
    "In the first step, using the 'naïve'γ5 prescription, which does not take into account terms with an odd number of γ5 matrices in fermion traces, we obtain the following differences between our incomplete expressions for the four-loop gauge beta functions ˆβi and the results of Refs. [32, 33]: ... However, if internal vertices are chosen as the reading points and the results are summed over all possible choices and divided by the number of internal vertices (in our case, three internal vertices for each fermion trace, giving an overall factor of 3 × 3 = 9), we exactly recover the excess contrib"

    Eqs. (26)-(28) tabulate the mismatch between the naive direct calculation and the WCC beta functions of Refs. [32,33]. The paper then chooses the internal-vertex/3x3-average reading point from among 'different results depending on the choice of reading point' precisely because it reproduces this mismatch. Therefore the four-loop beta functions 'fully agree with Refs. [32,33]' by construction for the gamma5-sensitive pieces, not by independent diagrammatic confirmation. The new gauge-field anomalous dimensions (33)-(34) are computed in that same calibrated scheme and lack an external cross-check; the argument that the 3x3 average is the physical limit is delegated to Ref. [25] and to WCCs which are themselves the source of the target result.

  2. other [Section 1 (Introduction) vs Section 3]
    "As for the treatment of chiral traces, we employ the reading point prescription for γ5 [40], following Ref. [25]. We find that the final results are independent of the auxiliary choices introduced by this procedure. ... As in previous analogous calculations [25], we obtain different results depending on the choice of reading point..."

    The Introduction claims independence of auxiliary choices, while Section 3 explicitly reports scheme dependence. The only reading-point choice that removes the difference from [32,33] is the internal-vertex average, so the 'independence' is not demonstrated; a single calibrated choice is imposed instead. This reinforces that the gamma5-sensitive agreement is achieved by selection rather than by an independent uniqueness argument.

full rationale

This is a large direct four-loop diagrammatic calculation, and substantial parts of the result—including the gauge-only and even-gamma5 terms—are genuine independent content, not circular. There is no meaningful self-citation chain: Refs. [25] and [32,33] are not by the present authors. However, the paper's central verification step for the gamma5-sensitive contributions is circular: the incomplete naive result is compared with the known WCC beta functions, and a gamma5 reading-point scheme is selected because it exactly reproduces the difference. Consequently, the concluding statement that the beta functions 'fully agree with Refs. [32,33]' is guaranteed by construction for exactly the terms that motivated the scheme choice. The new anomalous dimensions are obtained in the same calibrated scheme, so their uniqueness is contingent on the unproved physicality of the 3x3 internal-vertex average. Thus the central claim contains partial circularity, though not total reduction of the calculation to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No numbers are fitted to data and no new entities are introduced. The central claim rests on standard QFT machinery plus one non-standard choice: the gamma5 reading-point prescription, which is fixed by requiring agreement with previously known WCC results rather than derived from first principles.

axioms (6)
  • standard math Background-field Ward identity: Z_gi = Z_Vi^{-1/2} (Eq. 15).
    Standard identity in background-field gauge, used to convert background-field renormalization constants into coupling renormalization constants; not proved in this paper.
  • standard math MS renormalization scheme and the RG equations (19)-(24) are valid.
    The extraction of beta functions and anomalous dimensions from the 1/epsilon pole in Eq. (17) relies on the standard MS-like scheme; all results are scheme-specific.
  • ad hoc to paper The internal-vertex reading-point prescription for gamma5 yields the physical four-loop result.
    Section 3: different reading-point choices give different results; the internal-vertex averaged choice is adopted because it exactly reproduces the known WCC beta functions in Eqs. (26)-(28). This is the load-bearing scheme assumption.
  • domain assumption Fermion triangles with three vector lines sum to zero for SM field content with Nc=3.
    Section 3 states such contributions 'add up to zero when we take into account all SM fields and put the number of colors to be Nc=3'; relies on cancellation of gauge anomalies.
  • domain assumption FORCER correctly reduces all required four-loop massless propagator integrals.
    The calculation depends on FORCER [50] for parametric reduction; the paper provides no independent verification of the integral reductions.
  • domain assumption All dimensionful parameters (masses) can be dropped in the unbroken phase without affecting dimensionless RG functions.
    Stated in Sections 1-2; standard for mass-independent MS-like renormalization, but an assumption that mass effects decouple from the gauge beta functions at this order.

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read the original abstract

We present the beta functions of the gauge couplings and the anomalous dimensions of the gauge fields in the unbroken phase of the Standard Model at four loops.

Figures

Figures reproduced from arXiv: 2607.21586 by B.A.Kniehl, M.A.Bezuglov, V.N.Velizhanin.

Figure 1
Figure 1. Figure 1: Typical Feynman diagrams with odd numbers of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Reference graph

Works this paper leans on

50 extracted references · 36 linked inside Pith

  1. [1]

    B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, mr: A C++ library for the matching and running of the Standard Model parameters, Comput. Phys. C ommun. 206 (2016) 84–96 [arXiv:1601.08143 [hep-ph]]

    Pith/arXiv arXiv 2016

  2. [2]

    Sirlin and R

    A. Sirlin and R. Zucchini, Dependence of the Higgs coupling ¯h MS(M) on mH and the possible onset of new physics, Nucl. Phys. B 266 (1986) 389–409

    1986

  3. [3]

    Hempfling and B

    R. Hempfling and B. A. Kniehl, Relation between the fermion pole mas s and MS Yukawa coupling in the standard model, Phys. Rev. D 51 (1995) 1386–1394 [arXiv:hep-ph/9408313 [hep-ph]]

    Pith/arXiv arXiv 1995

  4. [4]

    Bezrukov, M

    F. Bezrukov, M. Yu. Kalmykov, B. A. Kniehl, and M. Shaposhnikov , Higgs boson mass and new physics, JHEP 10 (2012) 140 [arXiv:1205.2893 [hep-ph]]

    Pith/arXiv arXiv 2012

  5. [5]

    Jegerlehner, M

    F. Jegerlehner, M. Yu. Kalmykov, and B. A. Kniehl, On the differen ce between the pole and the MS masses of the top quark at the electroweak scale, Phys. Lett. B 722 (2013) 123–129 [arXiv:1212.4319 [hep-ph]]

    Pith/arXiv arXiv 2013

  6. [6]

    B. A. Kniehl and O. L. Veretin, Two-loop electroweak threshold c orrections to the bottom and top Yukawa couplings, Nucl. Phys. B 885 (2014) 459–480 [erratum: Nucl. Phys. B 894 (2015) 56–57] [arXiv:1401.1844 [hep-ph]]

    Pith/arXiv arXiv 2014

  7. [7]

    B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, Two-loop electrowe ak threshold corrections in the Standard Model, Nucl. Phys. B 896 (2015) 19–51 [arXiv:1503.02138 [hep-ph]]. 14

    Pith/arXiv arXiv 2015

  8. [8]

    A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, O n the b-quark run- ning mass in QCD and the SM, Nucl. Phys. B 916 (2017) 463–483 [arXiv:1612.00660 [hep-ph]]

    Pith/arXiv arXiv 2017

  9. [9]

    Degrassi, S

    G. Degrassi, S. Di Vita, J. Elias-Mir´ o, J. R. Espinosa, G. F. Giudic e, G. Isidori, and A. Strumia, Higgs mass and vacuum stability in the Standard Mode l at NNLO, JHEP 08 (2012) 098 [arXiv:1205.6497 [hep-ph]]

    Pith/arXiv arXiv 2012

  10. [10]

    Buttazzo, G

    D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, and A. Strumia, Investigating the near-criticality of the Higgs boson, J HEP 12 (2013) 089 [arXiv:1307.3536 [hep-ph]]

    Pith/arXiv arXiv 2013

  11. [11]

    A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner and O. L. Veretin, S tability of the Electroweak Vacuum: Gauge Independence and Advanced Precisio n, Phys. Rev. Lett. 115 (2015) 201802 [arXiv:1507.08833 [hep-ph]]

    Pith/arXiv arXiv 2015

  12. [12]

    Aad et al

    G. Aad et al. (ATLAS Collaboration), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at th e LHC, Phys. Lett. B 716 (2012) 1–29 [arXiv:1207.7214 [hep-ex]]

    Pith/arXiv arXiv 2012

  13. [13]

    Chatrchyan et al

    S. Chatrchyan et al. (CMS Collaboration), Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30–61 [arXiv:1207.7235 [hep-ex]]

    Pith/arXiv arXiv 2012

  14. [14]

    M. E. Machacek and M. T. Vaughn, Two-loop renormalization gro up equations in a general quantum field theory (I). Wave function renormalization, Nucl. Phys. B 222 (1983) 83–103

    1983

  15. [15]

    M. E. Machacek and M. T. Vaughn, Two-loop renormalization gro up equations in a general quantum field theory (II). Yukawa couplings, Nucl. Phys . B 236 (1984) 221–232

    1984

  16. [16]

    M. E. Machacek and M. T. Vaughn, Two-loop renormalization gro up equations in a general quantum field theory (III). Scalar quartic couplings, Nu cl. Phys. B 249 (1985) 70–92

    1985

  17. [17]

    L. N. Mihaila, J. Salomon, and M. Steinhauser, Gauge coupling bet a functions in the standard model to three loops, Phys. Rev. Lett. 108 (2012) 151602 [arXiv:1201.5868 [hep-ph]]

    Pith/arXiv arXiv 2012

  18. [18]

    L. N. Mihaila, J. Salomon, and M. Steinhauser, Renormalization co nstants and beta functions for the gauge couplings of the standard model to three -loop order, Phys. Rev. D 86 (2012) 096008 [arXiv:1208.3357 [hep-ph]]

    Pith/arXiv arXiv 2012

  19. [19]

    A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Mod el at three loops, JHEP 01 (2013) 017 [arXiv:1210.6873 [hep-ph]]

    Pith/arXiv arXiv 2013

  20. [20]

    K. G. Chetyrkin and M. F. Zoller, Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the standard model, JHEP 06 (2012) 033 [arXiv:1205.2892 [hep-ph]]. 15

    Pith/arXiv arXiv 2012

  21. [21]

    A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Yukawa co upling beta- functions in the Standard Model at three loops, Phys. Lett. B 722 (2013) 336–340 [arXiv:1212.6829 [hep-ph]]

    Pith/arXiv arXiv 2013

  22. [22]

    K. G. Chetyrkin and M. F. Zoller, β-function for the Higgs self-interaction in the Standard Model at three-loop level, JHEP 04 (2013)91 [erratum: JHEP 09 (2013) 155] [arXiv:1303.2890 [hep-ph]]

    Pith/arXiv arXiv 2013

  23. [23]

    A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Higgs self- coupling beta- function in the Standard Model at three loops, Nucl. Phys. B 875 (2013) 552–565 [arXiv:1303.4364 [hep-ph]]

    Pith/arXiv arXiv 2013

  24. [24]

    Jegerlehner, Facts of life with γ5, Eur

    F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673–679 [arXiv:hep- th/0005255 [hep-th]]

    arXiv 2001

  25. [25]

    A. V. Bednyakov and A. F. Pikelner, Four-loop strong coupling b eta-function in the Standard Model, Phys. Lett. B 762 (2016) 151–156 [arXiv:1508.02680 [hep-ph]]

    Pith/arXiv arXiv 2016

  26. [26]

    M. F. Zoller, Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level, JHEP 02 (2016) 095 [arXiv:1508.03624 [hep-ph]]

    Pith/arXiv arXiv 2016

  27. [27]

    Poole and A

    C. Poole and A. E. Thomsen, Weyl Consistency Conditions and γ5, Phys. Rev. Lett. 123 (2019) 041602 [arXiv:1901.02749 [hep-th]]

    Pith/arXiv arXiv 2019

  28. [28]

    Osborn, Weyl consistency conditions and a local renormaliza tion group equation for general renormalizable field theories, Nucl

    H. Osborn, Weyl consistency conditions and a local renormaliza tion group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486–526

    1991

  29. [29]

    Antipin, M

    O. Antipin, M. Gillioz, J. Krog, E. Mølgaard, and F. Sannino, Stand ard model vac- uum stability and Weyl consistency conditions, JHEP 08 (2013) 034 [arXiv:1306.3234 [hep-ph]]

    Pith/arXiv arXiv 2013

  30. [30]

    Jack and H

    I. Jack and H. Osborn, Constraints on RG flow for four dimensio nal quantum field theories, Nucl. Phys. B 883 (2014) 425–500 [arXiv:1312.0428 [hep-th]]

    Pith/arXiv arXiv 2014

  31. [31]

    Poole and A

    C. Poole and A. E. Thomsen, Constraints on 3- and 4-loop β-functions in a general four-dimensional Quantum Field Theory, JHEP 09 (2019) 055 [arXiv:1906.04625 [hep-th]]

    Pith/arXiv arXiv 2019

  32. [32]

    Davies, F

    J. Davies, F. Herren, C. Poole, M. Steinhauser, and A. E. Thom sen, Gauge Coupling β Functions to Four-Loop Order in the Standard Model, Phys. Rev. L ett. 124 (2020) 071803 [arXiv:1912.07624 [hep-ph]]

    Pith/arXiv arXiv 2020

  33. [33]

    Bednyakov and A

    A. Bednyakov and A. Pikelner, Four-Loop Gauge and Three-Lo op Yukawa Beta Functions in a General Renormalizable Theory, Phys. Rev. Lett. 127 (2021) 041801 [arXiv:2105.09918 [hep-ph]]

    Pith/arXiv arXiv 2021

  34. [34]

    Steudtner, General scalar renormalisation group equation s at three-loop order, JHEP 12 (2020) 012 [arXiv:2007.06591 [hep-th]]

    T. Steudtner, General scalar renormalisation group equation s at three-loop order, JHEP 12 (2020) 012 [arXiv:2007.06591 [hep-th]]

    Pith/arXiv arXiv 2020

  35. [35]

    Steudtner, Towards general scalar-Yukawa renormalisat ion group equations at three-loop order, JHEP 05 (2021) 060 [arXiv:2101.05823 [hep-th]]

    T. Steudtner, Towards general scalar-Yukawa renormalisat ion group equations at three-loop order, JHEP 05 (2021) 060 [arXiv:2101.05823 [hep-th]]. 16

    Pith/arXiv arXiv 2021

  36. [36]

    Davies, F

    J. Davies, F. Herren, and A. E. Thomsen, General gauge-Yuk awa-quartic β-functions at 4-3-2-loop order, JHEP 01 (2022) 051 [arXiv:2110.05496 [hep-ph]]

    Pith/arXiv arXiv 2022

  37. [37]

    Steudtner and A

    T. Steudtner and A. E. Thomsen, General quartic β-function at three loops, JHEP 10 (2024) 163 [arXiv:2408.05267 [hep-ph]]

    Pith/arXiv arXiv 2024

  38. [38]

    L. F. Abbott, The background field method beyond one loop, Nu cl. Phys. B 185 (1981) 189–203

    1981

  39. [39]

    L. F. Abbott, Introduction to the Background Field Method, A cta Phys. Polon. B 13 (1982) 33–50

    1982

  40. [40]

    J. G. K¨ orner, D. Kreimer, and K. Schilcher, A Practicable γ5-scheme in dimensional regularization, Z. Phys. C 54 (1992) 503–512

    1992

  41. [41]

    Denner, G

    A. Denner, G. Weiglein, and S. Dittmaier, Application of the backg round-field method to the electroweak standard model, Nucl. Phys. B 440 (1995) 95–128 [arXiv:hep-ph/9410338 [hep-ph]]

    Pith/arXiv arXiv 1995

  42. [42]

    S. A. Larin and J. A. M. Vermaseren, The three-loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334–336 [arXiv:hep-ph/9302208 [hep-ph]]

    Pith/arXiv arXiv 1993

  43. [43]

    O. V. Tarasov and A. A. Vladimirov, Yad. Fiz. 25 (1977) 1104–1106 [Sov. J. Nucl. Phys. 25 (1977) 585]

    1977

  44. [44]

    A. A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210–217 [Theor. Math. Phys. 43 (1980) 417–422]

    1980

  45. [45]

    O. V. Tarasov, A. A. Vladimirov, and A. Yu. Zharkov, The Gell-Ma nn-Low function of QCD in the three-loop approximation, Phys. Lett. B 93 (1980) 429–432

    1980

  46. [46]

    Tentyukov and J

    M. Tentyukov and J. Fleischer, A Feynman diagram analyzer DIA NA, Comput. Phys. Commun. 132 (2000) 124–141 [arXiv:hep-ph/9904258 [hep-ph]]

    Pith/arXiv arXiv 2000

  47. [47]

    Nogueira, Automatic Feynman Graph Generation, J

    P. Nogueira, Automatic Feynman Graph Generation, J. Comput . Phys. 105 (1993) 279–289

    1993

  48. [48]

    J. A. M. Vermaseren, New features of FORM, [arXiv:math-ph/0 010025 [math-ph]]

  49. [49]

    van Ritbergen, A

    T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41–96 [arXiv:hep-ph/9802376 [hep-ph]]

    Pith/arXiv arXiv 1999

  50. [50]

    Ruijl, T

    B. Ruijl, T. Ueda, and J. A. M. Vermaseren, FORCER, a FORM pro gram for the parametric reduction of four-loop massless propagator diagr ams, Comput. Phys. Commun. 253 (2020) 107198 [arXiv:1704.06650 [hep-ph]]. 17

    Pith/arXiv arXiv 2020