Pith. sign in

REVIEW 2 major objections 5 minor 91 references

Planar two-loop integrals for massive radiative return admit differential equations polynomial in ε that integrate stably in the physical region.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-12 03:08 UTC pith:QHRXPOKQ

load-bearing objection Solid, usable planar two-loop bases for massive radiative-return kinematics; the elliptic and nested-root constructions are careful and the numerics are honestly benchmarked. the 2 major comments →

arxiv 2607.03343 v1 pith:QHRXPOKQ submitted 2026-07-03 hep-ph hep-th

First look at the evaluation of two-loop Feynman integrals for radiative return processes

Mattia Pozzoli , William J. Torres Bobadilla This is my paper
classification hep-ph hep-th PACS 12.20.Ds13.40.Ks13.66.Bc
keywords radiative returntwo-loop Feynman integralsdifferential equationselliptic integralsnested square rootelectron massNNLO QEDmaster integrals
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.

Low-energy e⁺e⁻ experiments that use radiative return to measure hadronic vacuum polarisation need NNLO QED predictions that keep the full electron-mass dependence. The bottleneck is two-loop four-point integrals with massive propagators. This paper evaluates the three planar integral families that enter the initial-state radiation contribution to e⁺e⁻→γγ*. Despite elliptic curves and a nested square root, the authors construct bases of master integrals whose differential equations stay polynomial in the dimensional regulator ε and contain only algebraic coefficients. Numerical integration of those equations, starting from a carefully chosen physical boundary point, remains stable across the s-channel phase space of BaBar, Belle II, BESIII and KLOE kinematics, even though m_e²/s is of order 10⁻⁷–10⁻⁸. The results are the first concrete building blocks for a complete NNLO calculation of radiative-return processes with massive electrons.

Core claim

The planar two-loop four-point families PL1–PL3 admit bases of master integrals whose differential equations are polynomial in ε (at most quadratic) and free of nested square roots; these equations can be integrated numerically to stable results throughout the physical s-channel region relevant for low-energy radiative-return experiments, despite large mass hierarchies.

What carries the argument

Bases of master integrals chosen so that the connection matrix of the differential equations is of the form ∑_k ε^k (c_kα dlog W_α + d_kβ ω_β), with algebraic letters and one-forms only; elliptic sectors are treated via differentials of the first, second and third kind, and the nested-square-root sector is left linear in ε rather than fully ε-factorised.

Load-bearing premise

That keeping only terms up to order ε^4 in the master-integral expansion is enough for the finite part of the full amplitude, an assumption checked so far only for the closed-fermion-loop subset.

What would settle it

A direct numerical comparison, at a physical phase-space point, of the finite part of a complete two-loop amplitude assembled from these masters against an independent evaluation that retains higher powers of ε or uses a fully canonical elliptic basis; disagreement beyond the claimed digits would falsify the truncation and stability claims.

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

If this is right

  • The same bases and numerical pipeline can be reused for the non-planar families once they are reduced.
  • Initial-state radiation NNLO predictions for e⁺e⁻→π⁺π⁻γ and μ⁺μ⁻γ become feasible with full electron-mass dependence.
  • Monte-Carlo generators for radiative-return experiments can incorporate these masters via grids of boundary points for faster evaluation.
  • The construction shows that polynomial-in-ε equations remain practical for genuine four-point elliptic integrals with nested square roots.

Where Pith is reading between the lines

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

  • If the same strategy extends without new geometries, the full set of two-loop masters for massive e⁺e⁻→γγ* can be completed with only modest additional analytic work.
  • The observed numerical stiffness near thresholds suggests that a graded basis isolating non-logarithmic one-forms at O(ε^4) would further improve Monte-Carlo performance.
  • The same elliptic curves reappear in other massive four-point processes, so the one-forms catalogued here may be reusable beyond radiative return.

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

2 major / 5 minor

Summary. The paper computes the planar two-loop four-point integral families PL1–PL3 that enter initial-state radiation contributions to e^{+}e^{-}→γγ* (and related radiative-return processes) with full electron-mass dependence. Using IBP reduction (cross-checked with LiteRed/FiniteFlow, Kira-3 and NeatIBP) the authors construct bases of master integrals whose differential equations are polynomial in ε (at most quadratic) and free of nested square roots, while remaining algebraic in the kinematics. Elliptic sectors (five non-isomorphic curves) and one nested-square-root kite are treated by maximal-cut Baikov analysis, reducible super-sectors and Picard–Fuchs operators; the resulting DEs are integrated numerically in Julia from AMFlow boundary conditions and shown to be stable throughout the physical s-channel region despite m_e^{2}/s hierarchies of order 10^{-7}–10^{-8}. Ancillary files on Zenodo supply the complete bases, alphabets and one-forms.

Significance. If the results hold, the planar building blocks required for NNLO QED radiative-return amplitudes with massive electrons become available for the first time. The work supplies concrete, publicly released master-integral bases and a practical numerical pipeline that already achieves millisecond-scale evaluations on Phokhara phase-space points. The transparent treatment of elliptic kite sectors and the deliberate avoidance of nested roots for numerical work are reusable methodological contributions. The calculation is first-principles (no fitted parameters) and the numerical validation against two independent public codes is documented honestly.

major comments (2)
  1. §5, paragraph after Eq. (5.1) and the discussion of w_max=4: the truncation of the master-integral ε-expansion at O(ε⁴) is motivated solely by a preliminary study of the n_F fermion-loop subset of the amplitude. Because the bases are not canonical, higher-weight terms could in principle enter the finite remainder of the full amplitude. A short explicit statement of the highest weight that appears in a complete (even if partial) amplitude reconstruction, or a clear caveat that the truncation remains an assumption pending the full amplitude, would strengthen the claim that the published bases are immediately usable for NNLO phenomenology.
  2. §5, Fig. 7 and surrounding text: for families PL2 and PL3 the worst-case agreement with DiffExp drops to only two significant figures near thresholds in double precision. While the authors correctly flag this limitation, the central claim of “stable numerical evaluations throughout the physical region” rests on these numbers. Either a higher-precision demonstration (or a graded basis that postpones non-logarithmic forms to O(ε⁴)) should be supplied, or the abstract/conclusion wording should be qualified to “stable to a few digits in double precision, with further optimisation possible.”
minor comments (5)
  1. Table 1: the last two rows report “Elliptic sectors (MIs)” but do not list which sectors; a short footnote or reference to the figures in §4 would help the reader.
  2. Eq. (4.15) and the subsequent nested-root discussion: the polynomial Q₆ is deferred to the ancillary files; a brief indication of its degree and the kinematic variables it depends on would improve readability of the main text.
  3. Fig. 6 caption: the electron-mass dependence is set to zero for visualisation; this should be stated more prominently so that readers do not misinterpret the plotted physical region.
  4. References [10–13] are listed as arXiv preprints with 2026 dates; if they have since been published or updated, the citations should be refreshed.
  5. §4.3, after Eq. (4.20): the phrase “we then verify that the transformation yields ε-factorised DEs” is repeated almost verbatim a few lines later; a single statement would suffice.

Circularity Check

1 steps flagged

No significant circularity: first-principles IBP reduction, Baikov/Picard-Fuchs basis construction, and independent AMFlow/DiffExp numerical validation of planar families PL1–PL3.

specific steps
  1. self citation load bearing [§3 (strategy paragraph) and §5 (numerical method)]
    "Motivated by earlier works [48, 49], we aim to obtain DEs whose connection matrix can be expressed as … we integrate the DEs employing the strategy developed in [61] with an in-house implementation in the programming language Julia"

    The general form of the target DEs (polynomial in ε, algebraic one-forms) and the numerical integrator are taken from papers that share an author (Pozzoli in [49], Torres Bobadilla in [61]). These citations supply the methodological template rather than an independent external theorem; however they are not used to force the concrete bases or the numerical values of the PL1–PL3 integrals, so the circularity remains minor and non-load-bearing for the paper’s strongest claim.

full rationale

The paper’s load-bearing chain is the explicit construction of master-integral bases for the three planar families such that the connection matrices are polynomial in ε (at most quadratic) and free of nested square roots, followed by numerical integration of those DEs. Bases are obtained by maximal-cut Baikov analysis that identifies elliptic differentials of the first/second/third kind, by derivatives, by reducible super-sectors, and by Picard-Fuchs operators; the resulting DEs are then verified to have the claimed ε-structure both on and beyond the cut. Boundary values are supplied by the independent public package AMFlow; numerical solutions are cross-checked against DiffExp and against AMFlow at physical points. Self-citations ([48,49,61] and related methodological works) supply the general strategy of seeking polynomial-in-ε DEs and the Julia integration framework, but do not supply the target integrals, the specific bases, or the numerical results for PL1–PL3. No parameter is fitted to data and then re-presented as a prediction; no uniqueness theorem is imported from the authors’ own prior work to forbid alternatives; the nested-root sector is deliberately left non-canonical precisely to avoid introducing structures that would make the numerics circular or unstable. The only mild self-referential element is the methodological scaffolding, which is not load-bearing for the concrete claims. Hence the circularity score is 1.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The work rests entirely on standard multi-loop technology (IBP, differential equations, Baikov representation, Picard-Fuchs operators) plus the kinematic definition of the physical region. No free parameters are fitted; no new physical entities are postulated. The only modelling choices are the truncation order in ε and the decision to keep nested roots out of the DE matrices for numerical convenience.

axioms (4)
  • standard math Integration-by-parts identities reduce all integrals of a given family to a finite basis of master integrals.
    Invoked throughout §2–3; standard since Chetyrkin–Tkachov and Laporta.
  • domain assumption The physical s-channel region is defined by the positivity/negativity conditions on Mandelstam invariants and Gram determinants given in eqs. (2.5).
    Used to select the phase-space points for numerical validation (§5).
  • ad hoc to paper Truncation of the ε-expansion of master integrals at O(ε^4) captures the finite part of the amplitude.
    Stated in §5 on the basis of a preliminary n_F study; not proven for the full amplitude.
  • domain assumption Elliptic sectors can be treated by associating masters with differentials of the first, second and third kind without introducing periods into the DE matrices.
    Method of [40] applied in §4; standard in the elliptic-integral literature.

pith-pipeline@v1.1.0-grok45 · 30729 in / 2547 out tokens · 22646 ms · 2026-07-12T03:08:36.900642+00:00 · methodology

0 comments
read the original abstract

Precision studies of radiative return processes at low-energy electron--positron colliders require next-to-next-to-leading order QED predictions retaining full dependence on the electron mass. We present the calculation of planar two-loop four-point Feynman integrals relevant for initial-state radiation contributions to these processes. The calculation presents considerable analytical complexity, due to the presence of a nested square root and of integrals associated with elliptic geometries. We construct differential equations for the Feynman integrals which are polynomial in the dimensional regulator, and are suitable for numerical integration. We demonstrate stable numerical evaluations throughout the physical region relevant for low-energy experiments, despite the presence of large hierarchies of scales. Our results provide essential building blocks for NNLO predictions for radiative return processes.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

91 extracted references · 1 canonical work pages

  1. [1]

    Aliberti et al.,The anomalous magnetic moment of the muon in the Standard Model: an update,Phys

    R. Aliberti et al.,The anomalous magnetic moment of the muon in the Standard Model: an update,Phys. Rept.1143(2025) 1–158, [2505.21476]

    Pith/arXiv arXiv 2025

  2. [2]

    Aliberti et al.,Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections ine +e− collisions,2410.22882

    R. Aliberti et al.,Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections ine +e− collisions,2410.22882

    Pith/arXiv arXiv

  3. [3]

    Budassi, C

    E. Budassi, C. M. Carloni Calame, M. Ghilardi, A. Gurgone, G. Montagna, M. Moretti, O. Nicrosini, F. Piccinini and F. P. Ucci,Radiative return at NLOPS accuracy,2601.19530

    Pith/arXiv arXiv

  4. [4]

    Petit Ros` as, O

    P. Petit Ros` as, O. Shekhovtsova and W. J. Torres Bobadilla,Radiative return meets GVMD,2603.13171

    arXiv

  5. [5]

    C. M. Carloni Calame, M. Ghilardi, A. Gurgone, G. Montagna, M. Moretti, O. Nicrosini, F. Piccinini and F. P. Ucci,Structure-dependent radiative corrections toe +e− →π +π−γin the GVMD approach,2603.28621

    Pith/arXiv arXiv

  6. [6]

    T. Dave, J. Paltrinieri, P. Petit Ros` as and W. J. Torres Bobadilla,Tensor decomposition of e+e− →π +π−γto higher orders in the dimensional regulator,2604.16251

    Pith/arXiv arXiv

  7. [7]

    Badger, J

    S. Badger, J. Kry´ s, R. Moodie and S. Zoia,Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED,JHEP11(2023) 041, [2307.03098]

    Pith/arXiv arXiv 2023

  8. [8]

    V. S. Fadin and R. N. Lee,Two-loop radiative corrections toe +e− →γγ∗cross section, JHEP11(2023) 148, [2308.09479]

    Pith/arXiv arXiv 2023

  9. [9]

    F. V. Tkachov,A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions,Phys. Lett.100B(1981) 65–68

    1981

  10. [10]

    K. G. Chetyrkin and F. V. Tkachov,Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops,Nucl. Phys. B192(1981) 159–204

    1981

  11. [11]

    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]

    Pith/arXiv arXiv 2000

  12. [12]

    Barucchi and G

    G. Barucchi and G. Ponzano,Differential equations for one-loop generalized Feynman integrals,J. Math. Phys.14(1973) 396–401

    1973

  13. [13]

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

    1991

  14. [14]

    A. V. Kotikov,Differential equations method: The Calculation of vertex type Feynman diagrams,Phys. Lett. B259(1991) 314–322

    1991

  15. [15]

    Gehrmann and E

    T. Gehrmann and E. Remiddi,Differential equations for two loop four point functions, Nucl. Phys. B580(2000) 485–518, [hep-ph/9912329]

    Pith/arXiv arXiv 2000

  16. [16]

    Z. Bern, L. J. Dixon and D. A. Kosower,Dimensionally regulated pentagon integrals,Nucl. Phys. B412(1994) 751–816, [hep-ph/9306240]

    Pith/arXiv arXiv 1994

  17. [17]

    J. L. Bourjaily et al.,Functions Beyond Multiple Polylogarithms for Precision Collider Physics, inSnowmass 2021, 3, 2022.2203.07088. – 26 –

    arXiv 2021

  18. [18]

    Bargiela, H

    P. Bargiela, H. Frellesvig, R. Marzucca, R. Morales, F. Seefeld, M. Wilhelm and T.-Z. Yang,The spectrum of Feynman-integral geometries at two loops,JHEP05(2026) 057, [2512.13794]

    Pith/arXiv arXiv 2026

  19. [19]

    von Manteuffel and R

    A. von Manteuffel and R. M. Schabinger,A novel approach to integration by parts reduction,Phys. Lett. B744(2015) 101–104, [1406.4513]

    Pith/arXiv arXiv 2015

  20. [20]

    Peraro,Scattering amplitudes over finite fields and multivariate functional reconstruction,JHEP12(2016) 030, [1608.01902]

    T. Peraro,Scattering amplitudes over finite fields and multivariate functional reconstruction,JHEP12(2016) 030, [1608.01902]

    Pith/arXiv arXiv 2016

  21. [21]

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

    Pith/arXiv arXiv 2013

  22. [22]

    Adams, E

    L. Adams, E. Chaubey and S. Weinzierl,Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms,Phys. Rev. Lett.118(2017) 141602, [1702.04279]

    Pith/arXiv arXiv 2017

  23. [23]

    Adams and S

    L. Adams and S. Weinzierl,Theε-form of the differential equations for Feynman integrals in the elliptic case,Phys. Lett. B781(2018) 270–278, [1802.05020]

    Pith/arXiv arXiv 2018

  24. [24]

    Frellesvig,On epsilon factorized differential equations for elliptic Feynman integrals, JHEP03(2022) 079, [2110.07968]

    H. Frellesvig,On epsilon factorized differential equations for elliptic Feynman integrals, JHEP03(2022) 079, [2110.07968]

    Pith/arXiv arXiv 2022

  25. [25]

    Dlapa, J

    C. Dlapa, J. M. Henn and F. J. Wagner,An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals,JHEP08(2023) 120, [2211.16357]

    Pith/arXiv arXiv 2023

  26. [26]

    P¨ ogel, X

    S. P¨ ogel, X. Wang and S. Weinzierl,Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral,Phys. Rev. Lett.130(2023) 101601, [2211.04292]

    Pith/arXiv arXiv 2023

  27. [27]

    P¨ ogel, X

    S. P¨ ogel, X. Wang and S. Weinzierl,Bananas of equal mass: any loop, any order in the dimensional regularisation parameter,JHEP04(2023) 117, [2212.08908]

    Pith/arXiv arXiv 2023

  28. [28]

    Frellesvig and S

    H. Frellesvig and S. Weinzierl,Onε-factorised bases and pure Feynman integrals,SciPost Phys.16(2024) 150, [2301.02264]

    Pith/arXiv arXiv 2024

  29. [29]

    Driesse, G

    M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer and J. Usovitsch,Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature641(2025) 603–607, [2411.11846]

    Pith/arXiv arXiv 2025

  30. [30]

    C. Duhr, F. Porkert and S. F. Stawinski,Canonical differential equations beyond genus one, JHEP02(2025) 014, [2412.02300]

    arXiv 2025

  31. [31]

    C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi and F. J. Wagner,Aspects of canonical differential equations for Calabi-Yau geometries and beyond,JHEP06(2025) 128, [2503.20655]

    Pith/arXiv arXiv 2025

  32. [32]

    J. Chen, L. L. Yang and Y. Zhang,On an approach to canonicalizing elliptic Feynman integrals,JHEP04(2026) 077, [2503.23720]

    arXiv 2026

  33. [33]

    G¨ orges, C

    L. G¨ orges, C. Nega, L. Tancredi and F. J. Wagner,On a procedure to deriveϵ-factorised differential equations beyond polylogarithms,JHEP07(2023) 206, [2305.14090]. [41]ε-collaborationcollaboration, I. Bree et al.,Geometric Bookkeeping Guide to Feynman Integral Reduction andϵ-Factorized Differential Equations,Phys. Rev. Lett.136(2026) 241602, [2506.09124]...

    Pith/arXiv arXiv 2023

  34. [34]

    Chaubey and V

    E. Chaubey and V. Sotnikov,Elliptic Leading Singularities and Canonical Integrands,Phys. Rev. Lett.135(2025) 101903, [2504.20897]

    Pith/arXiv arXiv 2025

  35. [35]

    Forner, C

    F. Forner, C. C. Mella, C. Nega, L. Tancredi and F. J. Wagner,Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms,2604.25270

    Pith/arXiv arXiv

  36. [36]

    Remiddi,Differential equations for the two loop equal mass sunrise,Acta Phys

    E. Remiddi,Differential equations for the two loop equal mass sunrise,Acta Phys. Polon. B 34(2003) 5311–5322, [hep-ph/0310332]

    Pith/arXiv arXiv 2003

  37. [37]

    Laporta and E

    S. Laporta and E. Remiddi,Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B704(2005) 349–386, [hep-ph/0406160]

    Pith/arXiv arXiv 2005

  38. [38]

    Pozzorini and E

    S. Pozzorini and E. Remiddi,Precise numerical evaluation of the two loop sunrise graph master integrals in the equal mass case,Comput. Phys. Commun.175(2006) 381–387, [hep-ph/0505041]

    Pith/arXiv arXiv 2006

  39. [39]

    Badger, M

    S. Badger, M. Becchetti, N. Giraudo and S. Zoia,Two-loop integrals fort t+jet production at hadron colliders in the leading colour approximation,JHEP07(2024) 073, [2404.12325]

    Pith/arXiv arXiv 2024

  40. [40]

    Becchetti, D

    M. Becchetti, D. Canko, V. Chestnov, T. Peraro, M. Pozzoli and S. Zoia,Two-loop Feynman integrals for leading colourt tWproduction at hadron colliders,JHEP07(2025) 001, [2504.13011]

    Pith/arXiv arXiv 2025

  41. [41]

    Febres Cordero, G

    F. Febres Cordero, G. Figueiredo, M. Kraus, B. Page and L. Reina,Two-loop master integrals for leading-colorpp→t tHamplitudes with a light-quark loop,JHEP07(2024) 084, [2312.08131]

    Pith/arXiv arXiv 2024

  42. [42]

    Becchetti, C

    M. Becchetti, C. Dlapa and S. Zoia,Canonical differential equations for the elliptic two-loop five-point integral family relevant to tt¯+jet production at leading color,Phys. Rev. D112 (2025) L031501, [2503.03603]

    Pith/arXiv arXiv 2025

  43. [43]

    Aliaj, G

    R. Aliaj, G. Dian and G. Papathanasiou,Novel cluster-algebraic letters for 5- and 6-point QCD processes,2603.16743

    arXiv

  44. [44]

    Li, R.-Y

    S.-X. Li, R.-Y. Zhang, X.-F. Wang, P.-F. Li, X.-J. Wei, Y. Wang, Y. Jiang and Q.-h. Wang, Planar master integrals for two-loop NLO electroweak light-fermion contributions to gg→ZH,2604.27314

    Pith/arXiv arXiv

  45. [45]

    Boughezal, M

    R. Boughezal, M. Czakon and T. Schutzmeier,NNLO fermionic corrections to the charm quark mass dependent matrix elements in ¯B→X sγ,JHEP09(2007) 072, [0707.3090]

    Pith/arXiv arXiv 2007

  46. [46]

    Czakon,Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD,Phys

    M. Czakon,Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD,Phys. Lett. B664(2008) 307–314, [0803.1400]

    Pith/arXiv arXiv 2008

  47. [47]

    M. K. Mandal and X. Zhao,Evaluating multi-loop Feynman integrals numerically through differential equations,JHEP03(2019) 190, [1812.03060]

    Pith/arXiv arXiv 2019

  48. [48]

    M. L. Czakon and M. Niggetiedt,Exact quark-mass dependence of the Higgs-gluon form factor at three loops in QCD,JHEP05(2020) 149, [2001.03008]

    Pith/arXiv arXiv 2020

  49. [49]

    Czakon, R

    M. Czakon, R. V. Harlander, J. Klappert and M. Niggetiedt,Exact Top-Quark Mass Dependence in Hadronic Higgs Production,Phys. Rev. Lett.127(2021) 162002, [2105.04436]

    Pith/arXiv arXiv 2021

  50. [50]

    Calisto, R

    F. Calisto, R. Moodie and S. Zoia,Learning Feynman integrals from differential equations with neural networks,JHEP07(2024) 124, [2312.02067]

    Pith/arXiv arXiv 2024

  51. [51]

    Haisch and M

    U. Haisch and M. Niggetiedt,Exact two-loop amplitudes for Higgs plus jet production with a cubic Higgs self-coupling,JHEP10(2024) 236, [2408.13186]. – 28 –

    Pith/arXiv arXiv 2024

  52. [52]

    Petit Ros` as and W

    P. Petit Ros` as and W. J. Torres Bobadilla,Fast evaluation of Feynman integrals for Monte Carlo generators,JHEP09(2025) 210, [2507.12548]

    Pith/arXiv arXiv 2025

  53. [53]

    Badger, C

    S. Badger, C. Brancaccio, M. Becchetti, M. Czakon, H. B. Hartanto, R. Poncelet and S. Zoia,Higher-order QCD corrections to top-quark pair production in association with a jet,2511.11431

    arXiv

  54. [54]

    Badger, M

    S. Badger, M. Becchetti, C. Brancaccio, M. Czakon, H. B. Hartanto, R. Poncelet and S. Zoia,Double virtual QCD corrections tot t+jet production at the LHC,JHEP05(2026) 044, [2511.11424]

    Pith/arXiv arXiv 2026

  55. [55]

    Czakon and L

    M. Czakon and L. Tancredi,Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries,2606.30354

    Pith/arXiv arXiv

  56. [56]

    Liu, Y.-Q

    X. Liu, Y.-Q. Ma and C.-Y. Wang,A Systematic and Efficient Method to Compute Multi-loop Master Integrals,Phys. Lett. B779(2018) 353–357, [1711.09572]

    Pith/arXiv arXiv 2018

  57. [57]

    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]

    Pith/arXiv arXiv 2023

  58. [58]

    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]

    Pith/arXiv arXiv 2021

  59. [59]

    First look at the evaluation of two-loop Feynman integrals for radiative return processes

    M. Pozzoli and W. J. Torres Bobadilla,Ancillary files for “First look at the evaluation of two-loop Feynman integrals for radiative return processes”, July, 2026. 10.5281/zenodo.20826750

    work page doi:10.5281/zenodo.20826750 2026

  60. [60]

    R. N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685

    Pith/arXiv arXiv

  61. [61]

    Peraro,FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs,JHEP07(2019) 031, [1905.08019]

    T. Peraro,FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs,JHEP07(2019) 031, [1905.08019]

    Pith/arXiv arXiv 2019

  62. [62]

    Pak,The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques,J

    A. Pak,The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques,J. Phys. Conf. Ser.368(2012) 012049, [1111.0868]

    Pith/arXiv arXiv 2012

  63. [63]

    Lange, J

    F. Lange, J. Usovitsch and Z. Wu,Kira 3: integral reduction with efficient seeding and optimized equation selection,Comput. Phys. Commun.322(2026) 109999, [2505.20197]

    Pith/arXiv arXiv 2026

  64. [64]

    Z. Wu, J. Boehm, R. Ma, H. Xu and Y. Zhang,NeatIBP 1.0, a package generating small-size integration-by-parts relations for Feynman integrals,Comput. Phys. Commun. 295(2024) 108999, [2305.08783]

    Pith/arXiv arXiv 2024

  65. [65]

    Argeri, S

    M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk, U. Schubert and L. Tancredi, Magnus and Dyson Series for Master Integrals,JHEP03(2014) 082, [1401.2979]

    Pith/arXiv arXiv 2014

  66. [66]

    Dlapa, X

    C. Dlapa, X. Li and Y. Zhang,Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight,JHEP07(2021) 227, [2103.04638]

    Pith/arXiv arXiv 2021

  67. [67]

    Flieger and W

    W. Flieger and W. J. Torres Bobadilla,Landau and leading singularities in arbitrary space-time dimensions,Eur. Phys. J. Plus139(2024) 1022, [2210.09872]

    Pith/arXiv arXiv 2024

  68. [68]

    J. Henn, B. Mistlberger, V. A. Smirnov and P. Wasser,Constructing d-log integrands and computing master integrals for three-loop four-particle scattering,JHEP04(2020) 167, [2002.09492]

    Pith/arXiv arXiv 2020

  69. [69]

    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]. – 29 –

    Pith/arXiv arXiv 2018

  70. [70]

    Adams, E

    L. Adams, E. Chaubey and S. Weinzierl,Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter,Phys. Rev. Lett.121(2018) 142001, [1804.11144]

    Pith/arXiv arXiv 2018

  71. [71]

    Adams, E

    L. Adams, E. Chaubey and S. Weinzierl,Analytic results for the planar double box integral relevant to top-pair production with a closed top loop,JHEP10(2018) 206, [1806.04981]

    Pith/arXiv arXiv 2018

  72. [72]

    Frellesvig,The loop-by-loop Baikov representation — Strategies and implementation, JHEP04(2025) 111, [2412.01804]

    H. Frellesvig,The loop-by-loop Baikov representation — Strategies and implementation, JHEP04(2025) 111, [2412.01804]

    Pith/arXiv arXiv 2025

  73. [73]

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

    Pith/arXiv arXiv 1997

  74. [74]

    P. A. Baikov,Explicit solutions of the three loop vacuum integral recurrence relations,Phys. Lett. B385(1996) 404–410, [hep-ph/9603267]

    Pith/arXiv arXiv 1996

  75. [75]

    Frellesvig and C

    H. Frellesvig and C. G. Papadopoulos,Cuts of Feynman Integrals in Baikov representation, JHEP04(2017) 083, [1701.07356]

    Pith/arXiv arXiv 2017

  76. [76]

    Lang,Elliptic functions

    S. Lang,Elliptic functions. Graduate texts in mathematics. Springer, New York, NY, 2 ed., May, 1987

    1987

  77. [77]

    Caron-Huot, M

    S. Caron-Huot, M. Correia and M. Giroux,Recursive Landau Analysis,Phys. Rev. Lett. 135(2025) 131603, [2406.05241]

    Pith/arXiv arXiv 2025

  78. [78]

    Correia, M

    M. Correia, M. Giroux and S. Mizera,SOFIA: Singularities of Feynman integrals automatized,Comput. Phys. Commun.320(2026) 109970, [2503.16601]

    arXiv 2026

  79. [79]

    M¨ uller-Stach, S

    S. M¨ uller-Stach, S. Weinzierl and R. Zayadeh,Picard-Fuchs equations for Feynman integrals,Commun. Math. Phys.326(2014) 237–249, [1212.4389]

    Pith/arXiv arXiv 2014

  80. [80]

    Jiang, J

    X. Jiang, J. Liu, X. Xu and L. L. Yang,Symbol letters of Feynman integrals from Gram determinants,Phys. Lett. B864(2025) 139443, [2401.07632]

    Pith/arXiv arXiv 2025

Showing first 80 references.