Pith. sign in

REVIEW 2 major objections 3 minor 3 cited by

Analytic two-loop amplitudes for $q\bar{q}\to \gamma \gamma$ and $gg \to \gamma \gamma$ mediated by a heavy-quark loop

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives a fully analytic representation of the two-loop heavy-quark diphoton amplitudes as iterated integrals over elliptic and polylogarithmic kernels, built on a canonical basis of 154 master integrals.

desk verdict First fully analytic two-loop diphoton amplitudes with heavy-quark mass dependence, with strong numerical checks; the main soft spot is the sketched proof of the 11 extra master-integral relations. read the letter →

arxiv 2502.00118 v1 pith:XMXSCKMG submitted 2025-01-31 hep-ph hep-th

Matteo Becchetti , Federico Coro , Christoph Nega , Lorenzo Tancredi , Fabian J. Wagner This is my paper · ORCID
classification hep-phhep-th
keywords diphotonproductionheavy-quarkloopellipticFeynmanintegralsepsilon-factorizeddifferentialequationsiteratedhelicityamplitudestop-quarkmasseffectsseriesexpansions
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

Two-photon production at hadron colliders receives a correction from heavy quarks circulating in the loops, and at two-loop order these corrections have resisted analytic treatment because the relevant Feynman integrals sit on an elliptic curve. The paper claims to remove that obstacle: it finds a canonical basis of master integrals in which the dependence on the dimensional regulator $\epsilon$ factors out of the differential equations, and it writes the helicity amplitudes as independent iterated integrals over mixed rational, algebraic, and elliptic kernels. The analytic form then reveals concrete structure: the poles of the amplitudes contain no elliptic kernels at all, and six of the seventeen elliptic letters, precisely those carrying two powers of the holomorphic period, cancel from the finite part. The same differential equations also yield double series expansions around large quark mass and around the threshold $s = 4m^2$, which the paper shows evaluate a full helicity coefficient in $0.02$--$0.07$ seconds while matching high-precision numerical benchmarks across the physical region. The authors state that the same machinery transfers to the heavy-quark dijet and photon-plus-jet amplitudes with only a minor generalization.

What carries the argument

The central object is an epsilon-factorized (canonical) basis $\vec{I}$ of 154 master integrals spanning the three planar and two non-planar integral families that carry the two-loop diagrams, satisfying $d\vec{I} = \epsilon \sum_{i=1}^{74} G_i\,\omega_i\, \vec{I}$. Two corner integrals of the non-planar family live on the elliptic curve $Y^2 = P_4(X)$ with $P_4(X) = (m^2 - X)(m^2 + s - X)\bigl(m^2(m^2 - 3s) - X(2m^2 + s) + X^2\bigr)$, and the transcendental input for the elliptic blocks is the pair of Frobenius periods $\varpi_0, \varpi_1$ together with the puncture function $G(s,t,m^2)$ defined in eq. (4.31), which encodes the extra elliptic singularities. The construction follows the procedure of ref. [55]: integrand analysis on the maximal cut identifies forms of the first, second, and third kind on the curve; the inverse Wronskian made from the periods renders the homogeneous equations unipotent; and successive rotations remove the remaining non-factorized entries. For numerics, the carrying mechanism is series expansion of the full amplitude --- not of individual functions --- around the two expansion points, with blow-up coordinates chosen so that all singular lines cross normally, and boundary constants fixed by regularity and by one-loop tadpole-bubble integrals. The same differential-equation system provides the proof of the eleven extra master-integral relations.

What would settle it

Take the claimed canonical basis, substitute it into the original differential equations at a generic phase-space point such as $(s/m^2, t/m^2) = (11/3, -5/2)$, and check that the rotated matrices are exactly $\epsilon$ times matrices whose entries are independent of $\epsilon$; any residual $\epsilon^0$ piece would falsify the canonical claim. A decisive and easier check is to evaluate the eleven extra relations of eqs. (4.36)--(4.37) at high precision with an independent numerical method (for instance auxiliary mass flow) at several phase-space points: if any of these relations fails to vanish to all orders in $\epsilon$, the reduction from 165 to 154 master integrals, and with it the analytic amplitude representation, is wrong.

Watch

Extended reading notes

Core claim

On the paper's own terms, the finding is that the two-loop heavy-quark-mediated diphoton amplitudes admit a complete analytic description in terms of independent iterated integrals, bypassing the semi-numerical methods used previously. Standard integration-by-parts reduction yields 165 master integrals, and the authors reduce this to 154 by discovering eleven extra relations that they prove to all orders in $\epsilon$ from the canonical system and, for two of them, by explicit nonlinear transformations in Feynman-parameter space. The system is then rotated into an epsilon-factorized form $d\vec{I} = \epsilon \sum_i G_i\,\omega_i\, \vec{I}$ whose alphabet has 74 differential forms (the 'letters', or integration kernels): 12 rational letters, 45 algebraic letters involving 16 square roots, and 17 elliptic letters, of which 4 are modular. In the physical amplitudes, all iterated integrals with elliptic kernels cancel from the poles, and from the finite part the six elliptic letters $M_4, E_4, E_6, E_{11}, E_{12}, E_{13}$ --- exactly those with two powers of the holomorphic period $\varpi_0$ in the numerator --- drop out, confirming a pattern first seen in two-point correlators. The paper further claims that double series expansions obtained from the differential equations, around $m^2 = \infty$ (in blow-up variables $x_1 = -t\,m^2/s^2$, $x_2 = s/(4m^2)$) and around $(s,t) = (4m^2, 0)$, converge beyond their naive radius, and that with the $t \leftrightarrow u$ symmetry they cover the physical region up to $s \approx 8m^2$ with per-helicity-coefficient evaluation times of $0.02$ to $0.07$ seconds.

Load-bearing premise

The load-bearing premise is that the canonical basis is genuinely epsilon-factorized for the whole system of 154 master integrals: the paper describes the construction using the algorithm of ref. [55] plus manual steps, but it does not display the explicit rotation matrix or give a complete proof of global epsilon-factorization, and if that construction failed anywhere, the analytic iterated-integral representation would be invalid even though the final amplitudes could still pass numerical checks.

Editorial extensions

If this is right

  • The heavy-quark part of NNLO diphoton production can be evaluated from fully analytic expressions, with a complete helicity coefficient computed in $0.02$--$0.07$ seconds at a single phase-space point.
  • The cancellation from the finite part of the six elliptic letters carrying $\varpi_0^2$ confirms for a four-point amplitude the simplification pattern first observed in two-point correlators, indicating that physical quantities are systematically simpler than their integral representations.
  • The eleven extra relations reduce the independent master integrals from 165 to 154 and, since they are proven to all orders in $\epsilon$ through the canonical differential equations, they can be relied on in any future use of these integral families.
  • Starting from the canonical system, series expansions can be produced at any regular or singular point of the differential equations; the authors state that a minor generalization of the present setup yields the analogous heavy-quark results for dijet and photon-plus-jet production.

Reading between the lines

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

  • Editorial inference: the demonstrated convergence beyond the naive radius suggests that a handful of expansion points --- large mass, the $s = 4m^2$ threshold, and the deferred $m^2 \to 0$ limit --- could cover the entire physical phase space analytically; completing the small-mass expansion would remove the last gap for fully analytic NNLO diphoton predictions.
  • Editorial inference: the pattern is stated at order $\epsilon^0$; a natural sharpening is to check whether the same six elliptic letters also drop at order $\epsilon$, which would matter for subleading pieces of higher-order predictions and would support the conjecture that physical observables only feel the elliptic geometry through letters whose numerators are free of $\varpi_0^2$.
  • Editorial inference: only two of the eleven extra master-integral relations are proven by explicit transformations of Feynman parameters, so the remaining nine point to unexplored nonlinear symmetries of the graph polynomials; a systematic algorithm for such symmetries could shrink the integral basis and reduce reduction costs for the dijet and photon-plus-jet analogues the paper plans.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper computes the two-loop helicity amplitudes for q qbar -> gamma gamma and gg -> gamma gamma mediated by heavy-quark loops. After generating diagrams and performing IBP reduction, the authors identify 165 master integrals and assert 11 additional relations among them, yielding 154 independent integrals. They construct an epsilon-factorized canonical basis for this system using the algorithm of ref. [55] and manual steps, expressing all master integrals in terms of Chen iterated integrals over 74 differential forms (12 rational, 45 algebraic, and 17 elliptic). They use this representation to exhibit cancellations of large classes of elliptic letters in the finite parts, and they provide series expansions around m^2 = infinity and s = 4 m^2 that combine to give fast numerical evaluation over a large region of phase space. The final helicity coefficients are numerically checked against AMFlow at six phase-space points and against an independent group at three points.

Significance. If the derivation is fully correct, this is a substantial advance: an analytic, two-scale two-loop amplitude with elliptic geometry expressed in terms of iterated integrals, together with an efficient numerical implementation based on series expansions. The paper ships ancillary files on Zenodo, making the results reproducible. The numerical checks are extensive and the agreement with an independent team is a strong endorsement. However, the central claim of a derived canonical basis and the independence of the 154-integral system rests on steps that are only summarized rather than fully proven, so the logical chain from IBP reduction to the analytic representation deserves careful scrutiny.

major comments (2)
  1. [Section 4.2, Eqs. (4.36)-(4.37)] The reduction from 165 to 154 master integrals via the 11 extra relations is load-bearing for the canonical differential equations in Eq. (4.35) and for the iterated-integral representation of the amplitudes. Only two of these relations are supported by explicit Feynman-parameter transformations (and one of those is only described as 'similar'), while the remaining nine are asserted to follow from the canonical DEs plus regularity conditions without showing the homogeneous linear combinations or the boundary analysis. The numerical checks in Table 3 are performed on the final helicity coefficients at six phase-space points and do not directly test these relations. I request that the authors either provide the complete proof of all 11 relations, or provide a direct high-precision numerical verification of each combination in Eqs. (4.36)-(4.37) at several phase-space points (e.g., using AMFlow) and make this verification available in the ancillary files.
  2. [Section 4.1, Eqs. (4.26)-(4.29) and Eq. (4.35)] The construction of the epsilon-factorized basis is described in words, with the starting basis for the elliptic top sector given explicitly, but the full rotation matrix T and the systematic verification that all non-epsilon-factorized entries are removed for the entire 154-integral system are not documented in the manuscript. While the canonical basis is provided in the ancillary files, the claim of having 'derived a canonical basis' would be considerably strengthened if the paper stated clearly where in the ancillary files the transformation is provided, and if an automated check (e.g., substituting the basis into the original DEs and confirming the epsilon-factorized form to a given order) were reported. As written, the reader cannot independently confirm this central step from the paper alone.
minor comments (3)
  1. [Section 5, point 4] There is a typo: 'substnatially' should be 'substantially'.
  2. [Section 1, paragraph 6] There is a typo: 'phtons' should be 'photons'.
  3. [Appendix C, root definitions] The definitions of r5 and r6 appear garbled in the text (e.g., 'r5 = p t + t(4m2)'); since these roots are stated not to be required, please clarify their intended expressions or remove them if they are unused.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytic amplitudes follow from Feynman rules, IBP reduction, and differential equations, with boundary conditions from analytic tadpole/bubble integrals or from independent AMFlow evaluations; self-citations are methodological only.

full rationale

The derivation chain is self-contained. The amplitudes are obtained from Feynman diagrams, projectors, IBP reduction, and differential equations, with no parameter fitted to the final result. The epsilon-factorized basis is constructed using the published general algorithm of ref. [55], which is a methodology citation rather than an input that encodes the target amplitudes; the basis itself is supplied in ancillary files and is in principle directly checkable. Boundary conditions for the large-mass expansion come from analytic tadpole and bubble integrals, and for the threshold expansion from either transport of those boundaries or AMFlow, an independent numerical method. The series expansions are validated against AMFlow evaluations of the unexpanded helicity coefficients (Table 3) and against an independent group's results [101], so the checks are external benchmarks, not fits. The 11 extra relations of Sec. 4.2 are presented as identities derivable from homogeneous canonical differential equations plus regularity conditions; only two are exhibited via Feynman-parameter transformations, and the boundary-zero argument for the remaining nine is sketched rather than fully shown. This is an omitted-proof/correctness concern, not a circular reduction, because the relations are not assumed as inputs but claimed as consequences of the differential equations. Self-citations to refs. [55] and [21] concern the construction method and previously known polylogarithmic candidates, respectively, and do not carry the central claim by themselves. No step was found in which a prediction is equivalent by construction to a fitted parameter, a definition, or a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new particles or forces. It rests on standard QFT machinery, the recent canonical-basis algorithm of ref. [55], and numerical boundary values from AMFlow. The 11 extra relations among master integrals are derived from differential equations and regularity, but they depend on unstated linear-independence assumptions.

assumptions (4)
  • domain assumption The algorithm of ref. [55] yields a fully epsilon-factorized canonical basis for this multi-scale elliptic system.
    Section 4.1 relies on this recent algorithm; its applicability to the complete 154-master system is not proven in the paper.
  • domain assumption The 74 differential forms in eq. (4.35) are linearly independent over the relevant kinematic space.
    Section 4.2 uses this linear independence to prove the 11 extra relations, but no explicit proof is given.
  • domain assumption The boundary constants for the master integrals are correctly determined, with elliptic-master boundaries at s=4m^2 taken from AMFlow.
    Section 7.2 uses AMFlow for a few elliptic integrals; accuracy is asserted through later comparisons.
  • domain assumption The IBP reductions performed with Reduze2 and KIRA are correct.
    Section 3 relies on these tools; no independent verification is provided in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Analytic two-loop amplitudes for $q\bar{q}\to \gamma \gamma$ and $gg \to \gamma \gamma$ mediated by a heavy-quark loop." pith.science (2026). https://pith.science/paper/XMXSCKMG

@misc{pith2026250200118,
  author       = {Pith},
  title        = {Pith review of: Analytic two-loop amplitudes for $q\barq\to \gamma \gamma$ and $gg \to \gamma \gamma$ mediated by a heavy-quark loop},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMXSCKMG}},
  note         = {Machine review of arXiv:2502.00118}
}
read the original abstract

We address the analytic computation of the two-loop scattering amplitudes for the production of two photons in parton-parton scattering, mediated by loops of heavy quarks. Due to the presence of integrals of elliptic type, both partonic channels have been previously computed using semi-numerical methods. In this paper, leveraging new advances in the theory of differential equations for elliptic Feynman integrals, we derive a canonical basis for all integrals involved and compute them in terms of independent iterated integrals over elliptic and polylogarithmic differential forms. We use this representation to showcase interesting cancellations in the physical expressions for the scattering amplitudes. Furthermore, we address their numerical evaluation by producing series expansion representations for the whole amplitudes, which we demonstrate to be fast and numerically reliable across a large region of the phase space.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. The multiloop sunset to all orders

    hep-th 2026-03 conditional novelty 8.0 of 10

    Multiloop sunset integrals in D=2 are expressed as convergent sums of symmetric polynomials in logarithms of mass ratios, with a dimension-raising operator that propagates the result to D=4-2ε.

  2. Degenerations of flat connections on Riemann surfaces

    hep-th 2026-07 accept novelty 7.0 of 10

    Enriquez and DHS kernels on genus-h surfaces close under non-separating degeneration to genus h-1 kernels with two punctures whose generators are Bernoulli series in the original Lie algebra elements.

  3. Two-loop QCD corrections to $ZH$ and off-shell $Z$ boson pair production in gluon fusion

    hep-ph 2025-09 conditional novelty 6.0 of 10

    The paper provides fast, analytic two-loop virtual QCD corrections for gluon-induced ZH and off-shell ZZ production with full top-quark mass dependence, validated against numerical results at the sub-percent level.

Reference graph

Works this paper leans on

101 extracted references · 3 canonical work pages · cited by 3 Pith papers

  1. [55]

    Görges, C

    L. Görges, C. Nega, L. Tancredi and F. J. Wagner,On a procedure to deriveϵ-factorised differential equations beyond polylogarithms, JHEP 07 (2023) 206, [2305.14090]

    arXiv 2023

  2. [1]

    Anastasiou, E

    C. Anastasiou, E. W. N. Glover and M. E. Tejeda-Yeomans,Two loop QED and QCD corrections to massless fermion boson scattering, Nucl. Phys. B 629 (2002) 255–289, [hep-ph/0201274]

    arXiv 2002

  3. [2]

    Z. Bern, A. De Freitas and L. J. Dixon,Two loop amplitudes for gluon fusion into two photons, JHEP 09 (2001) 037, [hep-ph/0109078]

    arXiv 2001

  4. [3]

    Caola, A

    F. Caola, A. Von Manteuffel and L. Tancredi,Diphoton Amplitudes in Three-Loop Quantum Chromodynamics, Phys. Rev. Lett.126 (2021) 112004, [2011.13946]

    arXiv 2021

  5. [4]

    Bargiela, F

    P. Bargiela, F. Caola, A. von Manteuffel and L. Tancredi,Three-loop helicity amplitudes for diphoton production in gluon fusion, JHEP 02 (2022) 153, [2111.13595]

    arXiv 2022

  6. [5]

    Caola, A

    F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel and L. Tancredi,Three-Loop Gluon Scattering in QCD and the Gluon Regge Trajectory, Phys. Rev. Lett.128 (2022) 212001, [2112.11097]

    arXiv 2022

  7. [6]

    Caola, A

    F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel and L. Tancredi,Three-loop helicity amplitudes for four-quark scattering in massless QCD, JHEP 10 (2021) 206, [2108.00055]

    arXiv 2021

  8. [7]

    Caola, A

    F. Caola, A. Chakraborty, G. Gambuti, A. von Manteuffel and L. Tancredi,Three-loop helicity amplitudes for quark-gluon scattering in QCD, JHEP 12 (2022) 082, [2207.03503]

    arXiv 2022

Show all 101 references
  1. [8]

    Bargiela, A

    P. Bargiela, A. Chakraborty and G. Gambuti,Three-loop helicity amplitudes for photon+jet production, Phys. Rev. D107 (2023) L051502, [2212.14069]

    2023 arXiv

  2. [9]

    E. E. Kummer,Über die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. reine ang. Mathematik21 (1840) 74–90; 193–225; 328–371

  3. [10]

    A. B. Goncharov,Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197–318

    1995

  4. [11]

    A. B. Goncharov,Multiple polylogarithms, cyclotomy and modular complexes, Math.Res.Lett. 5 (1998) 497–516, [1105.2076]

    1998 arXiv

  5. [12]

    Vollinga and S

    J. Vollinga and S. Weinzierl,Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177, [hep-ph/0410259]

    2005 arXiv

  6. [13]

    Remiddi and J

    E. Remiddi and J. A. M. Vermaseren,Harmonic polylogarithms, Int. J. Mod. Phys.A15 (2000) 725–754, [hep-ph/9905237]

    2000 arXiv

  7. [14]

    Agarwal, F

    B. Agarwal, F. Buccioni, A. von Manteuffel and L. Tancredi,Two-loop leading colour QCD corrections toq ¯q → γγg and qg → γγq, JHEP 04 (2021) 201, [2102.01820]. – 42 –

    2021 arXiv

  8. [15]

    Agarwal, F

    B. Agarwal, F. Buccioni, A. von Manteuffel and L. Tancredi,Two-Loop Helicity Amplitudes for Diphoton Plus Jet Production in Full Color, Phys. Rev. Lett.127 (2021) 262001, [2105.04585]

    2021 arXiv

  9. [16]

    Badger, C

    S. Badger, C. Brønnum-Hansen, D. Chicherin, T. Gehrmann, H. B. Hartanto, J. Henn et al.,Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders, JHEP 11 (2021) 083, [2106.08664]

    2021 arXiv

  10. [17]

    De Laurentis, H

    G. De Laurentis, H. Ita and V. Sotnikov,Double-virtual NNLO QCD corrections for five-parton scattering. II. The quark channels, Phys. Rev. D109 (2024) 094024, [2311.18752]

    2024 arXiv

  11. [18]

    De Laurentis, H

    G. De Laurentis, H. Ita, M. Klinkert and V. Sotnikov,Double-virtual NNLO QCD corrections for five-parton scattering. I. The gluon channel, Phys. Rev. D109 (2024) 094023, [2311.10086]

    2024 arXiv

  12. [19]

    Agarwal, F

    B. Agarwal, F. Buccioni, F. Devoto, G. Gambuti, A. von Manteuffel and L. Tancredi, Five-parton scattering in QCD at two loops, Phys. Rev. D109 (2024) 094025, [2311.09870]

    2024 arXiv

  13. [20]

    Maltoni, M

    F. Maltoni, M. K. Mandal and X. Zhao,Top-quark effects in diphoton production through gluon fusion at next-to-leading order in QCD, Phys. Rev. D100 (2019) 071501, [1812.08703]

    2019 arXiv

  14. [21]

    Becchetti, R

    M. Becchetti, R. Bonciani, L. Cieri, F. Coro and F. Ripani,Two-loop form factors for diphoton production in quark annihilation channel with heavy quark mass dependence, JHEP 12 (2023) 105, [2308.11412]

    2023 arXiv

  15. [22]

    Becchetti, R

    M. Becchetti, R. Bonciani, L. Cieri, F. Coro and F. Ripani,Full top-quark mass dependence in diphoton production at NNLO in QCD, Phys. Lett. B848 (2024) 138362, [2308.10885]

    2024 arXiv

  16. [23]

    Sabry,Fourth order spectral functions for the electron propagator, Nucl

    A. Sabry,Fourth order spectral functions for the electron propagator, Nucl. Phys. 33 (1962) 401–430

    1962

  17. [24]

    F. C. S. Brown and A. Levin,Multiple Elliptic Polylogarithms, 1110.6917

    arXiv

  18. [25]

    Broedel, C

    J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer,Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112, [1412.5535]

    2015 arXiv

  19. [26]

    Broedel, C

    J. Broedel, C. Duhr, F. Dulat and L. Tancredi,Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093, [1712.07089]

    2018 arXiv

  20. [27]

    Broedel, C

    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi,Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023, [1809.10698]

    2019 arXiv

  21. [28]

    Liu and Y.-Q

    X. Liu and Y.-Q. Ma,AMFlow: A Mathematica package for Feynman integrals computation via auxiliary mass flow, Comput. Phys. Commun.283 (2023) 108565, [2201.11669]

    2023 arXiv

  22. [29]

    Moriello,Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP 01 (2020) 150, [1907.13234]

    F. Moriello,Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP 01 (2020) 150, [1907.13234]

    2020 arXiv

  23. [30]

    Hidding,DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput

    M. Hidding,DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun.269 (2021) 108125, [2006.05510]

    2021 arXiv

  24. [31]

    R. M. Prisco, J. Ronca and F. Tramontano,LINE: Loop Integrals Numerical Evaluation, 2501.01943

    arXiv

  25. [32]

    Bonciani, V

    R. Bonciani, V. Del Duca, H. Frellesvig, M. Hidding, V. Hirschi, F. Moriello et al., – 43 – Next-to-leading-order QCD corrections to Higgs production in association with a jet, Phys. Lett. B 843 (2023) 137995, [2206.10490]

    2023 arXiv

  26. [33]

    C. Duhr, F. Gasparotto, C. Nega, L. Tancredi and S. Weinzierl,On the electron self-energy to three loops in QED, JHEP 11 (2024) 020, [2408.05154]

    2024 arXiv

  27. [34]

    Forner, C

    F. Forner, C. Nega and L. Tancredi,On the photon self-energy to three loops in QED, 2411.19042

  28. [35]

    Marzucca, A

    R. Marzucca, A. J. McLeod and C. Nega,Two-Loop Master Integrals for Mixed QCD-EW Corrections to gg → H Through O(ϵ2), 2501.14435

  29. [36]

    Abreu, M

    S. Abreu, M. Becchetti, C. Duhr and M. A. Ozcelik,Two-loop form factors for pseudo-scalar quarkonium production and decay, JHEP 02 (2023) 250, [2211.08838]

    2023 arXiv

  30. [37]

    Delto, C

    M. Delto, C. Duhr, L. Tancredi and Y. J. Zhu,Two-Loop QED Corrections to the Scattering of Four Massive Leptons, Phys. Rev. Lett.132 (2024) 231904, [2311.06385]

    2024 arXiv

  31. [38]

    A. V. Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B254 (1991) 158–164

    1991

  32. [39]

    Remiddi,Differential equations for Feynman graph amplitudes, Nuovo Cim

    E. Remiddi,Differential equations for Feynman graph amplitudes, Nuovo Cim. A110 (1997) 1435–1452, [hep-th/9711188]

    1997 arXiv

  33. [40]

    Gehrmann and E

    T. Gehrmann and E. Remiddi,Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485–518, [hep-ph/9912329]

    2000 arXiv

  34. [41]

    F. V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B100 (1981) 65–68

    1981

  35. [42]

    Chetyrkin and F

    K. Chetyrkin and F. Tkachov,Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B 192 (1981) 159–204

    1981

  36. [43]

    Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int

    S. Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15 (2000) 5087–5159, [hep-ph/0102033]

    2000 arXiv

  37. [44]

    Arkani-Hamed, J

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka,Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125, [1012.6032]

    2012 arXiv

  38. [45]

    J. M. Henn,Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601, [1304.1806]

    2013 arXiv

  39. [46]

    K. T. Chen,Iterated path integrals, Bull. Amer. Math. Soc.83 (1977) 831

    1977

  40. [47]

    R. N. Lee,Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108, [1411.0911]

    2015 arXiv

  41. [48]

    Gehrmann, A

    T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs,The two-loop master integrals for qq → V V, JHEP 06 (2014) 032, [1404.4853]

    2014 arXiv

  42. [49]

    Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput

    C. Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun.222 (2018) 295–312, [1705.06252]

    2018 arXiv

  43. [50]

    Dlapa, J

    C. Dlapa, J. Henn and K. Yan,Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, JHEP 05 (2020) 025, [2002.02340]

    2020 arXiv

  44. [51]

    R. N. Lee,Libra: A package for transformation of differential systems for multiloop integrals, Comput. Phys. Commun.267 (2021) 108058, [2012.00279]. – 44 –

    2021 arXiv

  45. [52]

    J. Henn, B. Mistlberger, V. A. Smirnov and P. Wasser,Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP 04 (2020) 167, [2002.09492]

    2020 arXiv

  46. [53]

    J. Chen, X. Jiang, X. Xu and L. L. Yang,Constructing canonical Feynman integrals with intersection theory, Phys. Lett. B814 (2021) 136085, [2008.03045]

    2021 arXiv

  47. [54]

    J. Chen, X. Jiang, C. Ma, X. Xu and L. L. Yang,Baikov representations, intersection theory, and canonical Feynman integrals, JHEP 07 (2022) 066, [2202.08127]

    2022 arXiv

  48. [56]

    C. Duhr, F. Porkert and S. F. Stawinski,Canonical Differential Equations Beyond Genus One, 2412.02300

  49. [57]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman,Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44 (1972) 189–213

    1972

  50. [58]

    Peraro and L

    T. Peraro and L. Tancredi,Physical projectors for multi-leg helicity amplitudes, JHEP 07 (2019) 114, [1906.03298]

    2019 arXiv

  51. [59]

    Peraro and L

    T. Peraro and L. Tancredi,Tensor decomposition for bosonic and fermionic scattering amplitudes, Phys. Rev. D103 (2021) 054042, [2012.00820]

    2021 arXiv

  52. [60]

    Nogueira,Automatic Feynman Graph Generation, J

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

    1993

  53. [61]

    J. A. M. Vermaseren,New features of FORM, math-ph/0010025

    arXiv

  54. [62]

    Kuipers, T

    J. Kuipers, T. Ueda, J. A. M. Vermaseren and J. Vollinga,FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453–1467, [1203.6543]

    2013 arXiv

  55. [63]

    Ruijl, T

    B. Ruijl, T. Ueda and J. Vermaseren,FORM version 4.2, 1707.06453

    arXiv

  56. [64]

    Studerus,Reduze – Feynman integral reduction in C++, Comput

    C. Studerus,Reduze – Feynman integral reduction in C++, Comput. Phys. Commun.181 (2010) 1293–1300, [0912.2546]

    2010 arXiv

  57. [65]

    von Manteuffel and C

    A. von Manteuffel and C. Studerus,Reduze 2 - Distributed Feynman Integral Reduction, 1201.4330

    arXiv

  58. [66]

    Maierhöfer, J

    P. Maierhöfer, J. Usovitsch and P. Uwer,Kira—A Feynman integral reduction program, Comput. Phys. Commun.230 (2018) 99–112, [1705.05610]

    2018 arXiv

  59. [67]

    Klappert and F

    J. Klappert and F. Lange,Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247 (2020) 106951, [1904.00009]

    2020 arXiv

  60. [68]

    Klappert, F

    J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch,Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun.266 (2021) 108024, [2008.06494]

    2021 arXiv

  61. [69]

    Klappert, S

    J. Klappert, S. Y. Klein and F. Lange,Interpolation of dense and sparse rational functions and other improvements in FireFly, Comput. Phys. Commun.264 (2021) 107968, [2004.01463]

    2021 arXiv

  62. [70]

    Primo and L

    A. Primo and L. Tancredi,On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94–116, [1610.08397]

    2017 arXiv

  63. [71]

    Primo and L

    A. Primo and L. Tancredi,Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B 921 (2017) 316–356, [1704.05465]. – 45 –

    2017 arXiv

  64. [72]

    P. A. Baikov,Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A389 (1997) 347–349, [hep-ph/9611449]

    1997 arXiv

  65. [73]

    P. A. Baikov,Explicit solutions of the three loop vacuum integral recurrence relations, Phys. Lett. B 385 (1996) 404–410, [hep-ph/9603267]

    1996 arXiv

  66. [74]

    Frellesvig and C

    H. Frellesvig and C. G. Papadopoulos,Cuts of Feynman Integrals in Baikov representation, JHEP 04 (2017) 083, [1701.07356]

    2017 arXiv

  67. [75]

    Calisto, R

    F. Calisto, R. Moodie and S. Zoia,Learning Feynman integrals from differential equations with neural networks, JHEP 07 (2024) 124, [2312.02067]

    2024 arXiv

  68. [76]

    Czakon and A

    M. Czakon and A. Mitov,Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result, Nucl. Phys. B 824 (2010) 111–135, [0811.4119]

    2010 arXiv

  69. [77]

    von Manteuffel and L

    A. von Manteuffel and L. Tancredi,A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127, [1701.05905]

    2017 arXiv

  70. [78]

    Ahmed, E

    T. Ahmed, E. Chaubey, M. Kaur and S. Maggio,Two-loop non-planar four-point topology with massive internal loop, JHEP 05 (2024) 064, [2402.07311]

    2024 arXiv

  71. [79]

    Caron-Huot and J

    S. Caron-Huot and J. M. Henn,Iterative structure of finite loop integrals, JHEP 06 (2014) 114, [1404.2922]

    2014 arXiv

  72. [80]

    Analytic two-loop amplitudes forq ¯q → γγ and gg → γγ mediated by a heavy-quark loop

    M. Becchetti, F. Coro, C. Nega, L. Tancredi and F. J. Wagner,Ancillary Files to "Analytic two-loop amplitudes forq ¯q → γγ and gg → γγ mediated by a heavy-quark loop", 2025. 10.5281/zenodo.14733100

    2025 doi

  73. [81]

    Melnikov, R

    K. Melnikov, R. Rietkerk, L. Tancredi and C. Wever,Triple-real contribution to the quark beam function in QCD at next-to-next-to-next-to-leading order, JHEP 06 (2019) 033, [1904.02433]

    2019 arXiv

  74. [82]

    Buccioni, P

    F. Buccioni, P. A. Kreer, X. Liu and L. Tancredi,One loop QCD corrections to gg→ ttH at O ϵ2 , JHEP 03 (2024) 093, [2312.10015]

    2024 arXiv

  75. [83]

    Wu and Y

    Z. Wu and Y. Zhang,A new method for finding more symmetry relations of Feynman integrals, 2406.20016

    arXiv

  76. [84]

    Bärnreuther, M

    P. Bärnreuther, M. Czakon and P. Fiedler,Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP 02 (2014) 078, [1312.6279]

    2014 arXiv

  77. [85]

    Catani,The Singular behavior of QCD amplitudes at two loop order, Phys

    S. Catani,The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B427 (1998) 161–171, [hep-ph/9802439]

    1998 arXiv

  78. [86]

    Becher and M

    T. Becher and M. Neubert,Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett.102 (2009) 162001, [0901.0722]

    2009 arXiv

  79. [87]

    Becher and M

    T. Becher and M. Neubert,Infrared singularities of QCD amplitudes with massive partons, Phys. Rev. D79 (2009) 125004, [0904.1021]

    2009 arXiv

  80. [88]

    Becher and M

    T. Becher and M. Neubert,On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081, [0903.1126]

    2009 arXiv

  81. [89]

    Gardi and L

    E. Gardi and L. Magnea,Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079, [0901.1091]

    2009 arXiv

  82. [90]

    Ferroglia, M

    A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories, JHEP 11 (2009) 062, [0908.3676]. – 46 –

    2009 arXiv

  83. [91]

    Niggetiedt and J

    M. Niggetiedt and J. Usovitsch,The Higgs-gluon form factor at three loops in QCD with three mass scales, JHEP 02 (2024) 087, [2312.05297]

    2024 arXiv

  84. [92]

    R. N. Lee and V. A. Stotsky,Master integrals for e+e− → 2γ process at large energies and angles, JHEP 12 (2024) 106, [2410.03336]

    2024 arXiv

  85. [93]

    V. A. Smirnov,Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1–262

    2002

  86. [94]

    Melnikov and M

    K. Melnikov and M. Dowling,Production of two Z-bosons in gluon fusion in the heavy top quark approximation, Phys. Lett. B744 (2015) 43–47, [1503.01274]

    2015 arXiv

  87. [95]

    Davies, G

    J. Davies, G. Mishima, M. Steinhauser and D. Wellmann,Double Higgs boson production at NLO in the high-energy limit: complete analytic results, JHEP 01 (2019) 176, [1811.05489]

    2019 arXiv

  88. [96]

    Davies, G

    J. Davies, G. Mishima and M. Steinhauser,Virtual corrections togg → ZH in the high-energy and large-mt limits, JHEP 03 (2021) 034, [2011.12314]

    2021 arXiv

  89. [97]

    Davies, G

    J. Davies, G. Mishima, M. Steinhauser and D. Wellmann,gg → ZZ: analytic two-loop results for the low- and high-energy regions, JHEP 04 (2020) 024, [2002.05558]

    2020 arXiv

  90. [98]

    Griffiths and J

    P. Griffiths and J. Harris,Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994, 10.1002/9781118032527

    1994 doi

  91. [99]

    Hironaka,Resolution of singularities of an algebraic variety over a field of characteristic zero

    H. Hironaka,Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2)79 (1964), 109–203; ibid. (2)79 (1964) 205–326

    1964

  92. [100]

    Duhr and F

    C. Duhr and F. Dulat,PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135, [1904.07279]

    2019 arXiv

  93. [101]

    Ahmed, A

    T. Ahmed, A. Chakraborty, E. Chaubey and M. Kaur,Two-loop helicity amplitudes for diphoton production with massive quark loop, To appear soon. – 47 –

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.