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Four-Loop Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ in the BMHV Scheme

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Pith's one-line read Complete 4-loop renormalisation of a chiral gauge theory in the BMHV scheme

desk verdict First four-loop BMHV calculation, credible and technically impressive, but the completeness claim rests on two asserted-vanishing Green functions that need explicit justification. read the letter →

arxiv 2506.12253 v1 pith:JPD2772V submitted 2025-06-13 hep-ph hep-th

Andreas von Manteuffel , Dominik Stöckinger , Matthias Wei{ss}wange This is my paper · ORCID
classification hep-phhep-th MSC 81T1581T18 PACS 11.10.Gh12.15.-y
keywords BMHVschemenon-anticommutinggamma5dimensionalregularizationchiralgaugetheoryrenormalizationsymmetry-restoringcountertermsfour-loopquantumactionprinciple
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 claims to complete the four-loop renormalisation of an Abelian chiral gauge theory in the Breitenlohner-Maison / 't Hooft-Veltman (BMHV) scheme, the only self-consistent treatment of the Dirac matrix $\gamma_5$ in dimensional regularisation. The BMHV scheme breaks gauge and BRST invariance, and the paper determines all counterterms needed to restore the symmetry at four loops, including the finite symmetry-restoring part $S^4_{\mathrm{fct}}$ in Eq. (4.31). It supplies explicit analytic coefficients in Appendix B and numerical values for the Standard Model fermionic sector in Eq. (4.34). If correct, this is the first four-loop BMHV calculation and shows the scheme remains practically feasible at high loop orders, supporting future precision electroweak computations.

What carries the argument

The central object is the local composite operator $\Delta = s_D(S_0 + S_{\mathrm{ct}})$, whose insertions into Green functions encode the BRST symmetry breaking of the regularised theory, following the quantum action principle. Three technical mechanisms carry the calculation: (1) a BMHV algebra implemented in FORM in which every $\gamma$-trace factorises as $\frac{1}{4} \mathrm{Tr}(\hat{\gamma}_{\nu_1}\cdots\hat{\gamma}_{\nu_m})\,\mathrm{Tr}(\bar{\gamma}_{\mu_1}\cdots\bar{\gamma}_{\mu_n}\Lambda)$, separating evanescent from 4-dimensional parts; (2) a tadpole decomposition with an auxiliary mass that maps all UV-divergent integrals to fully massive vacuum bubbles; (3) an orbit-partition tensor reduction that reduces high-rank vacuum tensor integrals to scalar master integrals handled by IBP reduction.

What would settle it

Compute, in the same setup, the two four-loop $\Delta$-inserted 1PI Green functions $i\Delta\cdot\Gamma|^4_{\psi\psi B c}$ and $i\Delta\cdot\Gamma|^4_{BBBB c}$ (Eqs. (4.28)-(4.29)) and inspect their pole structure and finite parts; any non-vanishing result invalidates Eq. (4.31). A cheaper cross-check is to reproduce the coefficients of Appendix B for a smaller hypercharge matrix such as the Standard Model assignments in Eq. (4.33), using an independent implementation of the BMHV algebra.

Watch

Extended reading notes

Core claim

Working with a non-anticommuting $\gamma_5$, the paper computes every power-counting divergent 1PI Green function required for a complete renormalisation: the standard functions $B_\mu B_\nu$, $\bar\psi\psi$, $\bar\psi B \psi$ and $BBBB$, plus the $\Delta$-inserted functions that capture the regularisation-induced BRST breaking through the quantum action principle. From these it derives the complete four-loop counterterm action, decomposed into a BRST-invariant divergent part, a BRST-breaking divergent part, and a finite symmetry-restoring part, Eq. (4.31). The finite part contains no field monomials beyond those already present at three loops, so the structure of the counterterm action stabilises at four loops; the coefficients are rational combinations of Riemann zeta values in the hypercharge matrix. Applied to the fermionic sector of the Standard Model, the paper gives the numerical coefficients in Eq. (4.34).

Load-bearing premise

The paper assumes, without showing the cancellation, that the two power-counting divergent $\Delta$-inserted Green functions in Eqs. (4.28) and (4.29) vanish identically; if either one is nonzero, the complete set of symmetry-restoring counterterms in Eq. (4.31) would be missing contributions.

Editorial extensions

If this is right

  • A full 4-loop renormalisation of an Abelian chiral gauge theory in the BMHV scheme exists, so the scheme can be used self-consistently at that order.
  • The finite symmetry-restoring counterterm action $S^4_{\mathrm{fct}}$ has the same structure as at three loops, so no new field monomials appear at four loops; the paper argues power counting and renormalisability keep this structure at higher orders.
  • Numerical coefficients for the Standard Model fermionic sector are now available and can be plugged directly into 4-loop electroweak computations.
  • The UV-divergent BRST-breaking contributions obtained from standard and from $\Delta$-inserted Green functions agree exactly, a nontrivial consistency check that supports the result.

Reading between the lines

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

  • If the vanishing of Eqs. (4.28)-(4.29) is confirmed analytically, symmetry restoration in Abelian chiral BMHV theories can be automated at any loop order using only the small set of Green functions listed in Sec. 4, because the fermionic breaking can always be assigned to bilinear terms.
  • The trace-factorisation identity (3.12) could be exported to generic $\gamma_5$ problems in other gauge groups or effective field theories, potentially simplifying higher-loop SMEFT computations.
  • The large numerical coefficients in Eq. (4.34) (e.g. $F^{4,\mathrm{SM}}_{BB}\approx 140$) suggest finite symmetry-restoring counterterms contribute noticeably to 4-loop electroweak observables; one could test this by comparing physical quantities computed with the BMHV scheme against anticommuting-$\gamma_5$ prescriptions where the latter are unambiguous.
  • The same computational pipeline (tadpole decomposition, orbit-partition tensor reduction, FORM-based BMHV algebra) should extend to non-Abelian chiral gauge theories at 4 loops, where additional 5-leg Green functions and ghost contributions appear, provided the required master integrals are available.
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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 / 5 minor

Summary. This paper presents a four-loop renormalisation of an Abelian chiral gauge theory with right-handed fermions regularised in the BMHV scheme. The authors compute all power-counting divergent standard 1PI Green functions (Sec. 4.1), the BRST breaking encoded in Delta-inserted Green functions (Sec. 4.2), and the finite symmetry-restoring counterterm action S^4_fct given in Eq. (4.31), with explicit coefficients in App. B and numerical values for the SM-like fermionic sector in Eq. (4.34). The computational setup (Sec. 3) is described in detail, including the BMHV algebra, gamma-trace factorisation, tadpole decomposition, and tensor reduction. The paper claims this is the first complete 4-loop BMHV calculation and demonstrates the practical feasibility of the scheme at this order.

Significance. If correct, this is a significant milestone: the first 4-loop BMHV renormalisation of a chiral gauge theory, with the full set of singular and finite symmetry-restoring counterterms provided in analytical form. The paper includes strong internal consistency checks: the 3-loop results are reproduced, vector-like QED is validated at 4-loop order, and the divergent BRST-breaking poles agree between the two independent calculations described in Secs. 4.1 and 4.2. The explicit analytical results in App. B and the ancillary file are valuable resources for future applications. However, the completeness of the main result depends on the asserted but not demonstrated vanishing of two Delta-inserted Green functions (Eqs. (4.28)-(4.29)), which currently leaves the central claim conditional.

major comments (2)
  1. [Sec. 4.2, Eqs. (4.28)-(4.29)] The two power-counting divergent Delta-inserted 1PI Green functions iDelta*Gamma|^4_{psi psi B c} and iDelta*Gamma|^4_{BBBB c} are asserted to vanish identically, with the only stated support being "a cancellation of the leading power-counting term in the respective integrands" and, for the latter, an appeal to renormalisability. No diagram sum, projector result, or code output is shown. Because these Green functions would produce the dimension-5 BRST-breaking monomials c psi psi B and c B^4 in Delta^4_ct, a non-vanishing value would directly modify the finite counterterm action S^4_fct in Eq. (4.31) and would invalidate the claimed completeness of the 4-loop renormalisation. The agreement of delta-X coefficients and the locality of the computed poles, used as consistency checks in Secs. 4.2-4.3, are independent of the vanishing and would not detect an error here. Please provide the explicit demonstration, ideally with the relevant integrals after projection, or a detailed derivation of the cancellation at the integrand level.
  2. [Sec. 4.2, Eq. (4.29)] The renormalisability argument invoked for iDelta*Gamma|^4_{BBBB c} = 0 is not self-evident. The operator cB^4 has canonical dimension 5, and the quantum action principle in DReg does not by itself forbid power-counting divergent insertions of such operators into the Slavnov-Taylor identity; the cited Refs. [25,33] treat lower orders (and different operator content), so they do not directly settle the 4-loop case. The argument should be spelled out explicitly, for instance by showing that no local counterterm of dimension <= 4 can absorb such a breaking, and it should be cross-checked against the explicit loop integrands.
minor comments (5)
  1. [Eq. (4.18)] The notation "LIM D -> 4" is non-standard; please define it as the limit D -> 4 after dropping evanescent terms, or replace it with a conventional limit symbol and a footnote.
  2. [Eqs. (4.23) and (4.30)] The symbol "O(hat.)" is undefined; please state explicitly that it denotes finite evanescent terms that vanish in the limit D -> 4, and explain the meaning of the dot.
  3. [Sec. 3.1] The validation against 4-loop vector-like QED is mentioned but no details or references are given; please add a reference to the QED calculation or specify which cross-checked quantities were compared.
  4. [Eq. (4.16)] The phrase "does also not introduce" should read "does not introduce".
  5. [Eq. (4.33)] The hypercharge matrix is printed as a column vector; since it is presumably diagonal, please display it as a diagonal matrix or state explicitly that the listed entries are the diagonal elements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 4-loop counterterms are computed from Feynman integrals with independent QED and lower-order cross-checks; the asserted vanishing of two Delta-inserted Green functions is a verification gap, not a circular reduction.

full rationale

The paper's main output, S^4_fct in Eq. (4.31), is not a fitted quantity or a restatement of an input. The coefficients F^4,break are extracted from explicit 4-loop evaluations of Delta-inserted 1PI Green functions, Eqs. (4.24)-(4.27), which are computed from Feynman rules using the BMHV algebra, tadpole decomposition, and tensor reduction described in Sec. 3. The only structural input is the quantum action principle of Breitenlohner-Maison and the tree-level breaking operator Delta; the perturbative recursion in Eq. (4.21) is standard and not circular because lower-order counterterms are outputs of previous orders rather than assumptions about the 4-loop result. Independent support is present: a complete 4-loop QED renormalisation validated against the literature, a reproduction of the previous 3-loop chiral results, and external master integrals and IBP tools (FIRE, Kira, Finred, Refs. [42-54]). The self-citations to Refs. [25,33] for the vanishing of the cB^4 Green function are not load-bearing circularity: the argument invoked is power-counting and renormalisability, is parameter-free, and the paper also cites an integrand-level cancellation for Eq. (4.28). The genuinely questionable point, Eqs. (4.28)-(4.29), is an unshown cancellation that would be a completeness or correctness gap if wrong, but it is not a reduction of the claimed result to its own inputs; it is an asserted computational fact. The non-uniqueness of the finite symmetry-restoring counterterms is explicitly documented in Sec. 4.3, and no fitted parameter is renamed as a prediction. Accordingly, no circular step is established.

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

The calculation has no fitted parameters; all counterterm coefficients are computed. The main input assumptions are the BMHV scheme definition, the quantum action principle, literature master integrals, and the asserted vanishing of two Green functions.

assumptions (5)
  • standard math Cyclicity of the Dirac trace in D dimensions, and the trace normalization Tr(1)=4.
    Used in the derivation of the trace factorization formula in App. A, Eq. (3.12).
  • domain assumption The BMHV algebra defined in Sec. 3.3, in particular the decomposition of gamma matrices into 4- and (D-4)-dimensional parts with the commutation relations (3.5)-(3.7).
    This is the definition of the regularization scheme; the entire calculation rests on these relations.
  • domain assumption The quantum action principle of DReg, i.e., S_D(Gamma_Ren) = Delta . Gamma_Ren, is valid.
    Used to derive the symmetry restoration condition in Eq. (4.19)-(4.21).
  • standard math The 4-loop master integrals from Ref. [54] are correct.
    Used in the final reduction of scalar integrals; the paper does not rederive them.
  • ad hoc to paper The two power-counting divergent Delta-inserted Green functions in Eqs. (4.28)-(4.29) vanish identically.
    Asserted without proof in Sec. 4.2; if false, the counterterm set is incomplete.

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Cite this review

Pith. "Pith review of Four-Loop Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ in the BMHV Scheme." pith.science (2026). https://pith.science/paper/JPD2772V

@misc{pith2026250612253,
  author       = {Pith},
  title        = {Pith review of: Four-Loop Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ in the BMHV Scheme},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPD2772V}},
  note         = {Machine review of arXiv:2506.12253}
}
abstract

We present the complete 4-loop renormalisation of an Abelian chiral gauge theory in the Breitenlohner-Maison / `t Hooft-Veltman (BMHV) scheme. Employing a non-anticommuting $\gamma_5$ in dimensional regularisation, we determine the full set of symmetry restoring counterterms from the quantum action principle. Our calculation represents the highest-order application of the BMHV framework so far, pushing the limits of a self-consistent treatment of $\gamma_5$ at the multi-loop level. We describe the computational setup that we developed to perform the computations and discuss key implementation aspects, such as the BMHV algebra and tensor reduction. Our work demonstrates the feasibility of applying the BMHV scheme at high loop orders and establishes a solid foundation for future studies of high-precision electroweak physics.

Figures

Figures reproduced from arXiv: 2506.12253 by the authors.

Figure 1
Figure 1. The 4-point Green function cBµBνBρ , while having a comparatively low degree of divergence of 1, involves a significantly larger number Feynman diagrams, see [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 1
Figure 1. Representative 4-loop Feynman diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Representative 4-loop Feynman diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: Representative 3-loop Feynman diagrams with counterterm insertions contribut [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]

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Reference graph

Works this paper leans on

76 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Regularization and Renormalization of Gauge Fields,

    G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields,” Nucl. Phys. B , vol. 44, pp. 189–213, 1972.doi: 10.1016/0550-3213(72) 90279-9

    work page doi:10.1016/0550-3213(72 1972

  2. [2]

    Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 1.,

    P. Breitenlohner and D. Maison, “Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 1.,” Commun. Math. Phys. , vol. 52, p. 39,

    work page

  3. [3]

    Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 2.,

    P. Breitenlohner and D. Maison, “Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 2.,” Commun. Math. Phys. , vol. 52, p. 55,

    work page

  4. [4]

    Dimensional Renormalization and the Action Principle,

    P. Breitenlohner and D. Maison, “Dimensional Renormalization and the Action Principle,” Commun. Math. Phys.,vol.52,pp.11–38,1977. doi: 10.1007/BF01609069. 31

    doi:10.1007/bf01609069 1977

  5. [5]

    doi: 10.1007/BF01609071

    work page doi:10.1007/bf01609071

  6. [6]

    Introduction to Renormalization Theory and Chiral Gauge Theor- ies in Dimensional Regularization with Non-Anticommutingγ5,

    H. Bélusca-Maïto, A. Ilakovac, P. Kühler, M. Mađor-Božinović, D. Stöckinger and M. Weißwange, “Introduction to Renormalization Theory and Chiral Gauge Theor- ies in Dimensional Regularization with Non-Anticommutingγ5,” Symmetry, vol. 15, no. 3, p. 622, 2023.doi: 10.3390/sym15030622. arXiv: 2303.09120 [hep-ph]

    arXiv 2023

  7. [7]

    Facts of life with gamma(5),

    F. Jegerlehner, “Facts of life with gamma(5),”Eur. Phys. J. C , vol. 18, pp. 673–679,

    work page

  8. [8]

    A Practicable gamma(5) scheme in dimensional regularization,

    J. G. Korner, D. Kreimer and K. Schilcher, “A Practicable gamma(5) scheme in dimensional regularization,” Z. Phys. C , vol. 54, pp. 503–512, 1992.doi: 10.1007/ BF01559471

    work page 1992

Show all 76 references
  1. [9]

    The Role of gamma(5) in dimensional regularization,

    D. Kreimer, “The Role of gamma(5) in dimensional regularization,” Dec. 1993. arXiv: hep-ph/9401354

    1993 arXiv

  2. [10]

    The γ(5) Problem and Anomalies: A Clifford Algebra Approach,

    D. Kreimer, “The γ(5) Problem and Anomalies: A Clifford Algebra Approach,” Phys. Lett. B , vol. 237, pp. 59–62, 1990.doi: 10.1016/0370-2693(90)90461-E

    1990 doi

  3. [11]

    Three-loop \beta-functions for top-Yukawa and the Higgs self-interaction in the Standard Model,

    K. G. Chetyrkin and M. F. Zoller, “Three-loop \beta-functions for top-Yukawa and the Higgs self-interaction in the Standard Model,”JHEP, vol. 06, p. 033, 2012.doi: 10.1007/JHEP06(2012)033. arXiv: 1205.2892 [hep-ph]

    2012 arXiv

  4. [12]

    Top-Yukawa effects on theβ-function of the strong coupling in the SM at four-loop level,

    M. F. Zoller, “Top-Yukawa effects on theβ-function of the strong coupling in the SM at four-loop level,” JHEP, vol. 02, p. 095, 2016.doi: 10.1007/JHEP02(2016)095. arXiv: 1508.03624 [hep-ph]

    2016 arXiv

  5. [13]

    The Renormalization of the axial anomaly in dimensional regulariz- ation,

    S. A. Larin, “The Renormalization of the axial anomaly in dimensional regulariz- ation,” Phys. Lett. B , vol. 303, pp. 113–118, 1993.doi: 10.1016/0370-2693(93) 90053-K. arXiv: hep-ph/9302240

    1993 arXiv

  6. [14]

    Weyl Consistency Conditions andγ5,

    C. Poole and A. E. Thomsen, “Weyl Consistency Conditions andγ5,” Phys. Rev. Lett., vol. 123, no. 4, p. 041602, 2019.doi: 10.1103/PhysRevLett.123.041602 . arXiv: 1901.02749 [hep-th]

    2019 arXiv

  7. [15]

    GaugeCoupling β Functions to Four-Loop Order in the Standard Model,

    J.Davies,F.Herren,C.Poole,M.SteinhauserandA.E.Thomsen,“GaugeCoupling β Functions to Four-Loop Order in the Standard Model,”Phys. Rev. Lett., vol. 124, no. 7, p. 071803, 2020. doi: 10 . 1103 / PhysRevLett . 124 . 071803. arXiv: 1912 . 07624 [hep-ph]

    2020

  8. [16]

    Four-loop strong coupling beta-function in the Standard Model,

    A. V. Bednyakov and A. F. Pikelner, “Four-loop strong coupling beta-function in the Standard Model,” Phys. Lett. B , vol. 762, pp. 151–156, 2016.doi: 10.1016/j. physletb.2016.09.007. arXiv: 1508.02680 [hep-ph]

    2016 arXiv

  9. [17]

    Higher-order β-functions in the Standard Model and beyond,

    F. Herren, “Higher-order β-functions in the Standard Model and beyond,”SciPost Phys. Proc., vol. 7, p. 029, 2022.doi: 10.21468/SciPostPhysProc.7.029. arXiv: 2110.06938 [hep-ph]

    2022 arXiv

  10. [18]

    An observation on Feynman diagrams with axial anomalous subgraphs in dimensional regularization with an anticommutingγ5,

    L. Chen, “An observation on Feynman diagrams with axial anomalous subgraphs in dimensional regularization with an anticommutingγ5,” JHEP, vol. 2023, no. 11, p. 30, 2023.doi: 10.1007/JHEP11(2023)030. arXiv: 2304.13814 [hep-ph]

    2023 arXiv

  11. [19]

    Generalgauge-Yukawa-quartic β-functions at 4-3-2-loop order,

    J.Davies,F.HerrenandA.E.Thomsen,“Generalgauge-Yukawa-quartic β-functions at 4-3-2-loop order,”JHEP, vol. 01, p. 051, 2022.doi: 10.1007/JHEP01(2022)051. arXiv: 2110.05496 [hep-ph]

    2022 arXiv

  12. [20]

    Dimen- sional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme forγ5 ap- plied to chiral YM theories: full one-loop counterterm and RGE structure,

    H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, “Dimen- sional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme forγ5 ap- plied to chiral YM theories: full one-loop counterterm and RGE structure,”JHEP, vol. 08, no. 08, p. 024, 2020.doi: 10.1...

    2020 arXiv

  13. [21]

    Spurious gauge-invariance andγ5 in Dimensional Regularization,

    P. Olgoso Ruiz and L. Vecchi, “Spurious gauge-invariance andγ5 in Dimensional Regularization,” Jun. 2024. arXiv:2406.17013 [hep-ph]

    2024 arXiv

  14. [22]

    A Procedure g5anchor to Anchorγ5 in Feynman Diagrams for the Stand- ard Model,

    L. Chen, “A Procedure g5anchor to Anchorγ5 in Feynman Diagrams for the Stand- ard Model,” Sep. 2024. arXiv:2409.08099 [hep-ph]. 32

    2024 arXiv

  15. [23]

    Gauge invariance and finite counterterms in chiralgaugetheories,

    C. Cornella, F. Feruglio and L. Vecchi, “Gauge invariance and finite counterterms in chiralgaugetheories,” JHEP,vol.02,p.244,2023. doi: 10.1007/JHEP02(2023)244. arXiv: 2205.10381 [hep-ph]

    2023 arXiv

  16. [24]

    BRS symmetry restoration of chiral Abelian Higgs-Kibble theory in dimensional renormalization with a nonanticommuting gamma(5),

    D. Sanchez-Ruiz, “BRS symmetry restoration of chiral Abelian Higgs-Kibble theory in dimensional renormalization with a nonanticommuting gamma(5),”Phys. Rev. D, vol. 68, p. 025009, 2003. doi: 10 . 1103 / PhysRevD . 68 . 025009. arXiv: hep - th/0209023

    2003

  17. [25]

    C. P. Martin and D. Sanchez-Ruiz, “Action principles, restoration of BRS symmetry and the renormalization group equation for chiral nonAbelian gauge theories in dimensional renormalization with a nonanticommuting gamma(5),”Nucl. Phys. B , vol. 572, pp. 387–477, 2000.doi: 10.10...

  18. [26]

    Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme,

    L. Naterop and P. Stoffer, “Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme,”JHEP, vol. 02, p. 068, 2024.doi: 10.1007/JHEP02(2024)068. arXiv: 2310.13051 [hep-ph]

    2024 arXiv

  19. [27]

    γ5 schemes and the interplay of smeft operators in the higgs-gluon coupling,

    S. Di Noi, R. Gröber, G. Heinrich, J. Lang and M. Vitti, “ γ5 schemes and the interplay of smeft operators in the higgs-gluon coupling,”Phys. Rev. D , vol. 109, no. 9, p. 095024, 2024.doi: 10.1103/PhysRevD.109.095024 . arXiv: 2310.18221 [hep-ph]

    2024 arXiv

  20. [28]

    Shedding light on evan- escent shadows — Exploration of non-anticommutingγ5 in Dimensional Regular- isation,

    P. L. Ebert, P. Kühler, D. Stöckinger and M. Weißwange, “Shedding light on evan- escent shadows — Exploration of non-anticommutingγ5 in Dimensional Regular- isation,” JHEP, vol. 01, p. 114, 2025. doi: 10 . 1007 / JHEP01(2025 ) 114. arXiv: 2411.02543 [hep-ph]

    2025 arXiv

  21. [29]

    Renormalisation group effects in SMEFT for di-Higgs production,

    G. Heinrich and J. Lang, “Renormalisation group effects in SMEFT for di-Higgs production,” Sep. 2024. arXiv:2409.19578 [hep-ph]

    2024 arXiv

  22. [30]

    Renormalization-group equations of the LEFT at two loops: dimension-five effects,

    L. Naterop and P. Stoffer, “Renormalization-group equations of the LEFT at two loops: dimension-five effects,” Dec. 2024. arXiv:2412.13251 [hep-ph]

    2024 arXiv

  23. [31]

    Subleading operators and γ5-scheme dependence in SMEFT for Higgs boson pair production,

    G. Heinrich, S. Di Noi, R. Gröber, J. Lang, L. Scyboz and M. Vitti, “Subleading operators and γ5-scheme dependence in SMEFT for Higgs boson pair production,” PoS, vol. LL2024, p. 021, 2024.doi: 10.22323/1.467.0021 . arXiv: 2407.04747 [hep-ph]

    2024 arXiv

  24. [32]

    Renormalisation group equations for BRST-restored chiral the- ory in dimensional renormalisation: application to two-loop chiral-QED,

    H. Bélusca-Maïto, “Renormalisation group equations for BRST-restored chiral the- ory in dimensional renormalisation: application to two-loop chiral-QED,”JHEP, vol.03,p.202,2023. doi: 10.1007/JHEP03(2023)202.arXiv: 2208.09006 [hep-th]

    2023 arXiv

  25. [33]

    Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ5 in the BMHV scheme,

    D. Stöckinger and M. Weißwange, “Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ5 in the BMHV scheme,” JHEP, vol.02,p.139,2024. doi: 10.1007/JHEP02(2024)139.arXiv: 2312.11291 [hep-ph]

    2024 arXiv

  26. [34]

    Bélusca-Maïto, A

    H. Bélusca-Maïto, A. Ilakovac, P. Kühler, M. Mađor-Božinović and D. Stöckinger, “Two-loop application of the Breitenlohner-Maison/’t Hooft-Veltman scheme with non-anticommuting γ5: full renormalization and symmetry-restoring counterterms in an abelian chiral gauge theory,” JHE...

    2021 arXiv

  27. [35]

    Advances at theγ5-Frontier,

    P. Kühler, D. Stöckinger and M. Weißwange, “Advances at theγ5-Frontier,” PoS, vol. LL2024, p. 022, 2024. doi: 10 . 22323 / 1 . 467 . 0022. arXiv: 2407 . 07247 [hep-ph]

    2024

  28. [36]

    New features of FORM,

    J. A. M. Vermaseren, “New features of FORM,” Oct. 2000. arXiv: math - ph / 0010025

    2000

  29. [37]

    Two-Loop Renormalization of a ChiralSU (2) Gauge Theory in Dimensional Regularization with Non-Anticommuting γ5,

    P. Kühler and D. Stöckinger, “Two-Loop Renormalization of a ChiralSU (2) Gauge Theory in Dimensional Regularization with Non-Anticommuting γ5,” Apr. 2025. arXiv: 2504.06080 [hep-ph]

    2025 arXiv

  30. [38]

    FORM version 4.3.1 — Reference Manual

    B. Ruijl et al. “FORM version 4.3.1 — Reference Manual.” (Apr. 2023), [Online]. Available: file:///home/matthias/Downloads/form- 4.3.1- manual.pdf . (ac- cessed: 21.10.2024)

    2023

  31. [39]

    Automatic Feynman Graph Generation,

    P. Nogueira, “Automatic Feynman Graph Generation,”J. Comput. Phys. , vol. 105, pp. 279–289, 1993.doi: 10.1006/jcph.1993.1074

    1993

  32. [40]

    FORM version 4.2,

    B. Ruijl, T. Ueda and J. Vermaseren, “FORM version 4.2,” Jul. 2017. arXiv:1707. 06453 [hep-ph]

    2017

  33. [41]

    Semi- and Fully-Inclusive Phase-Space Integrals at Four Loops,

    V. Maheria, “Semi- and Fully-Inclusive Phase-Space Integrals at Four Loops,” Ph.D. dissertation, Hamburg U., 2022

    2022

  34. [42]

    FIRE6: Feynman Integral REduction with Modular Arithmetic,

    A. V. Smirnov and F. S. Chuharev, “FIRE6: Feynman Integral REduction with Modular Arithmetic,” Comput. Phys. Commun. , vol. 247, p. 106877, 2020. doi: 10.1016/j.cpc.2019.106877. arXiv: 1901.07808 [hep-ph]

    2020

  35. [43]

    Fully massive vacuum integrals at 5 loops,

    T. Luthe, “Fully massive vacuum integrals at 5 loops,” Ph.D. dissertation, Bielefeld U., 2015

    2015

  36. [44]

    Reduze 2 - Distributed Feynman Integral Reduction,

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

    2012 arXiv

  37. [45]

    Kira—A Feynman integral reduction program,

    P. Maierhöfer, J. Usovitsch and P. Uwer, “Kira—A Feynman integral reduction program,” Comput. Phys. Commun. , vol. 230, pp. 99–112, 2018.doi: 10.1016/j. cpc.2018.04.012. arXiv: 1705.05610 [hep-ph]

    2018 arXiv

  38. [46]

    FIRE 6.5: Feynman integral reduction with new simplification library,

    A. V. Smirnov and M. Zeng, “FIRE 6.5: Feynman integral reduction with new simplification library,” Comput. Phys. Commun. , vol. 302, p. 109261, 2024. doi: 10.1016/j.cpc.2024.109261. arXiv: 2311.02370 [hep-ph]

    2024

  39. [47]

    Reconstructing rational functions with FireFly,

    J. Klappert and F. Lange, “Reconstructing rational functions with FireFly,”Com- put. Phys. Commun., vol. 247, p. 106951, 2020.doi: 10.1016/j.cpc.2019.106951. arXiv: 1904.00009 [cs.SC]. 34

    2020

  40. [48]

    Interpolation of dense and sparse rational functions and other improvements in FireFly,

    J. Klappert, S. Y. Klein and F. Lange, “Interpolation of dense and sparse rational functions and other improvements in FireFly,”Comput. Phys. Commun. , vol. 264, p. 107968, 2021. doi: 10.1016/j.cpc.2021.107968. arXiv: 2004.01463 [cs.MS]

    2021

  41. [49]

    Integral reduction with Kira 2.0 and finite field methods,

    J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, “Integral reduction with Kira 2.0 and finite field methods,”Comput. Phys. Commun. , vol. 266, p. 108024,

  42. [50]

    Quark and gluon form factors to four- loop order in QCD: theN 3 f contributions,

    A. von Manteuffel and R. M. Schabinger, “Quark and gluon form factors to four- loop order in QCD: theN 3 f contributions,” Phys. Rev. D, vol. 95, no. 3, p. 034030,

  43. [51]

    Scattering amplitudes over finite fields and multivariate functional re- construction,

    T. Peraro, “Scattering amplitudes over finite fields and multivariate functional re- construction,” JHEP, vol. 12, p. 030, 2016.doi: 10.1007/JHEP12(2016)030. arXiv: 1608.01902 [hep-ph]

    2016 arXiv

  44. [52]

    High-precision epsilon expansions of single-mass- scale four-loop vacuum bubbles,

    Y. Schroder and A. Vuorinen, “High-precision epsilon expansions of single-mass- scale four-loop vacuum bubbles,”JHEP, vol. 06, p. 051, 2005.doi: 10.1088/1126- 6708/2005/06/051. arXiv: hep-ph/0503209

    2005 arXiv

  45. [53]

    A novel approach to integration by parts reduction,

    A. von Manteuffel and R. M. Schabinger, “A novel approach to integration by parts reduction,” Phys. Lett. B , vol. 744, pp. 101–104, 2015.doi: 10.1016/j.physletb. 2015.03.029. arXiv: 1406.4513 [hep-ph]

    2015 arXiv

  46. [54]

    The Four-loop QCD beta-function and anomalous dimensions,

    M. Czakon, “The Four-loop QCD beta-function and anomalous dimensions,”Nucl. Phys. B, vol. 710, pp. 485–498, 2005.doi: 10.1016/j.nuclphysb.2005.01.012 . arXiv: hep-ph/0411261

    2005 arXiv

  47. [55]

    Wolfram, Mathematica 12.0, Published by Wolfram Research Inc., 2019

    S. Wolfram, Mathematica 12.0, Published by Wolfram Research Inc., 2019. [Online]. Available: https://www.wolfram.com

    2019

  48. [56]

    Four-loop QCD β-function with different fermion representations of the gauge group,

    M. F. Zoller, “Four-loop QCD β-function with different fermion representations of the gauge group,” JHEP, vol. 10, p. 118, 2016.doi: 10.1007/JHEP10(2016)118 . arXiv: 1608.08982 [hep-ph]

    2016 arXiv

  49. [57]

    Four-loop renormalization of QCD with a redu- cible fermion representation of the gauge group: anomalous dimensions and renor- malization constants,

    K. G. Chetyrkin and M. F. Zoller, “Four-loop renormalization of QCD with a redu- cible fermion representation of the gauge group: anomalous dimensions and renor- malization constants,” JHEP, vol. 06, p. 074, 2017.doi: 10.1007/JHEP06(2017)

    2017 doi

  50. [58]

    Evaluation of the general 3-loop vacuum Feyn- man integral,

    S. P. Martin and D. G. Robertson, “Evaluation of the general 3-loop vacuum Feyn- man integral,” Phys. Rev. D , vol. 95, no. 1, p. 016008, 2017. doi: 10 . 1103 / PhysRevD.95.016008. arXiv: 1610.07720 [hep-ph]

    2017 arXiv

  51. [59]

    The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge,

    T. Luthe, A. Maier, P. Marquard and Y. Schroder, “The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge,” JHEP, vol. 10, p. 166, 2017.doi: 10.1007/JHEP10(2017)166. arXiv: 1709.07718 [hep-ph]

    2017 arXiv

  52. [60]

    Massive loop amplitudes from unitarity,

    Z. Bern and A. G. Morgan, “Massive loop amplitudes from unitarity,”Nucl. Phys. B, vol. 467, pp. 479–509, 1996. doi: 10 . 1016 / 0550 - 3213(96 ) 00078 - 8. arXiv: hep-ph/9511336

    1996 arXiv

  53. [61]

    Two loop helicity amplitudes for gluon- gluon scattering in QCD and supersymmetric Yang-Mills theory,

    Z. Bern, A. De Freitas and L. J. Dixon, “Two loop helicity amplitudes for gluon- gluon scattering in QCD and supersymmetric Yang-Mills theory,”JHEP, vol. 03, p. 018, 2002.doi: 10.1088/1126-6708/2002/03/018. arXiv: hep-ph/0201161. 35

    2002 arXiv

  54. [62]

    Mixed EW- QCD two-loop amplitudes forq ¯q → ℓ+ℓ− and γ5 scheme independence of multi-loop corrections,

    M. Heller, A. von Manteuffel, R. M. Schabinger and H. Spiesberger, “Mixed EW- QCD two-loop amplitudes forq ¯q → ℓ+ℓ− and γ5 scheme independence of multi-loop corrections,” JHEP, vol. 05, p. 213, 2021.doi: 10.1007/JHEP05(2021)213. arXiv: 2012.05918 [hep-ph]

    2021 arXiv

  55. [63]

    Two loop mixing of dimension five flavor changing op- erators,

    M. Misiak and M. Munz, “Two loop mixing of dimension five flavor changing op- erators,” Phys. Lett. B , vol. 344, pp. 308–318, 1995.doi: 10.1016/0370-2693(94) 01553-O. arXiv: hep-ph/9409454

    1995 arXiv

  56. [64]

    The five-loop beta function of Yang-Mills theory with fermions,

    F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, “The five-loop beta function of Yang-Mills theory with fermions,” JHEP, vol. 02, p. 090, 2017. doi: 10.1007/JHEP02(2017)090. arXiv: 1701.01404 [hep-ph]

    2017 arXiv

  57. [65]

    Two-Loop Rational Terms in Yang-Mills Theories,

    J.-N. Lang, S. Pozzorini, H. Zhang and M. F. Zoller, “Two-Loop Rational Terms in Yang-Mills Theories,”JHEP, vol. 10, p. 016, 2020.doi: 10.1007/JHEP10(2020)016. arXiv: 2007.03713 [hep-ph]

    2020 arXiv

  58. [66]

    The R∗-operation and five-loop calculations,

    B. Ruijl, F. Herzog, T. Ueda, J. A. M. Vermaseren and A. Vogt, “The R∗-operation and five-loop calculations,”PoS, vol. RADCOR2017, p. 011, 2018.doi: 10.22323/ 1.290.0011. arXiv: 1801.06084 [hep-ph]

    2018 arXiv

  59. [67]

    Tensor reduction of loop integrals,

    C. Anastasiou, J. Karlen and M. Vicini, “Tensor reduction of loop integrals,”JHEP, vol.12,p.169,2023. doi: 10.1007/JHEP12(2023)169.arXiv: 2308.14701 [hep-ph]

    2023 arXiv

  60. [68]

    Tensor reduction for Feynman integrals with Lorentz and spinor indices,

    J. Goode, F. Herzog, A. Kennedy, S. Teale and J. Vermaseren, “Tensor reduction for Feynman integrals with Lorentz and spinor indices,”JHEP, vol. 11, p. 123, 2024. doi: 10.1007/JHEP11(2024)123. arXiv: 2408.05137 [hep-ph]

    2024 arXiv

  61. [69]

    OPITeR: A program for tensor reduction of multi-loop Feynman Integrals,

    J. Goode, F. Herzog and S. Teale, “OPITeR: A program for tensor reduction of multi-loop Feynman Integrals,” Nov. 2024. arXiv:2411.02233 [hep-ph]

    2024 arXiv

  62. [70]

    Beta functions and anomalous di- mensions up to three loops,

    K. G. Chetyrkin, M. Misiak and M. Munz, “Beta functions and anomalous di- mensions up to three loops,” Nucl. Phys. B , vol. 518, pp. 473–494, 1998. doi: 10.1016/S0550-3213(98)00122-9. arXiv: hep-ph/9711266

    1998 arXiv

  63. [74]

    arXiv: 1704.04209 [hep-ph]

    arXiv

  64. [76]

    JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes,

    D. Binosi, J. Collins, C. Kaufhold and L. Theussl, “JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes,”Comput. Phys. Commun., vol. 180, pp. 1709–1715, 2009. doi: 10 . 1016 / j . cpc . 2009 . 02 . 020. arXiv: 0811.4113 [hep-ph]. 36

    2009 arXiv

  65. [1977]

    doi: 10.1007/BF01609070

    doi

  66. [2001]

    arXiv: hep-th/0005255

    doi: 10.1007/s100520100573. arXiv: hep-th/0005255

    arXiv

  67. [2017]

    arXiv: 1611.00795 [hep-ph]

    doi: 10.1103/PhysRevD.95.034030. arXiv: 1611.00795 [hep-ph]

    arXiv

  68. [2021]

    arXiv: 2008.06494 [hep-ph]

    doi: 10.1016/j.cpc.2021.108024. arXiv: 2008.06494 [hep-ph]

    2021

Pith tools

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