Pith. sign in

REVIEW 4 major objections 6 minor 83 references

Breaking dual conformal invariance turns the nine-particle N=4 super Yang-Mills alphabet into the first candidate symbol letters for six-point one-mass QCD integrals, including nested square roots and predictions beyond two loops.

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 · deepseek-v4-flash

2026-08-02 17:58 UTC pith:UTLW7Q6F

load-bearing objection A useful, honest paper that produces the first candidate alphabets for 6-point one-mass and new massless/2-mass QCD integrals from cluster algebras; the central prediction is conditional on a conjectural equivalence the authors themselves flag, and the 162/168 count slip should be fixed. the 4 major comments →

arxiv 2603.16743 v2 pith:UTLW7Q6F submitted 2026-03-17 hep-th hep-ph

Novel cluster-algebraic letters for 5- and 6-point QCD processes

Rigers Aliaj , Gabriele Dian , Georgios Papathanasiou This is my paper
classification hep-th hep-ph
keywords cluster algebrassymbol alphabetsdual conformal invarianceN=4 super Yang-MillsQCD scattering amplitudesnested square rootsmulti-loop Feynman integralsmomentum twistors
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish that the cluster-algebraic symbol alphabet proposed for nine-particle planar N=4 super Yang-Mills amplitudes can be carried over to QCD: by representing massive legs as pairs of massless ones and sending a dual point to infinity — breaking dual conformal invariance — the same kinds of letters describe planar QCD Feynman integrals at five and six points. The headline results are the first candidate alphabets for six-point integrals with one massive leg, roughly 245 letters of which 29 are genuinely six-point; among these are nested square-root letters whose radicands contain further square roots, a feature previously tied to more complicated, non-polylogarithmic integral families. As checks, the prediction essentially contains the complete one-loop six-point one-mass alphabet and, in the massless limit, essentially the full finite two-loop six-point massless alphabet, while adding 168 letters (162 in the abstract) not seen before, plus 87 new five-point two-mass letters. A sympathetic reader would care because the method turns a symmetry argument into concrete, testable input for multi-loop QCD calculations at colliders.

Core claim

The paper's central claim is that the cluster-algebraic alphabet proposed for nine-particle N=4 super Yang-Mills scattering survives the breaking of dual conformal invariance and thereby predicts the symbol alphabets of planar QCD integrals. Reducing the 9-point alphabet to subalphabets with massive legs and taking a dual point to infinity produces roughly 245 candidate letters for six-point one-mass processes: 119 rational, 59 rationalisable, and 68 with non-rationalisable square roots. Of the 29 genuinely six-point letters, twelve are nested square-root letters (Ai ± Biϵ ± Ci√Δ±)/(Ai ± Biϵ ∓ Ci√Δ±) whose radicands Δ± = F ± Gϵ contain a further square root in the Mandelstam variables. The m

What carries the argument

The engine is 'breaking dual conformal invariance': massive external legs are written as sums of massless ones in momentum twistor space, and one dual coordinate is sent to infinity so that dual-conformally-invariant cross ratios collapse to ordinary Lorentz-invariant Mandelstam variables while the letter alphabet is preserved. Subalphabets for (9−k)-point k-mass kinematics are extracted as nullspaces of annihilation operators O_{i,j} — the momentum-twistor form of BCFW shifts — applied to the 9-particle cluster alphabet, and solved with finite-field arithmetic. The nested square roots emerge because the genuinely six-point radicands Δ± are parity conjugates containing the rationalisable pse

Load-bearing premise

The entire chain stands or falls on whether the proposed nine-particle N=4 super Yang-Mills alphabet is complete and on whether breaking dual conformal invariance preserves the alphabet of every dual-conformally-invariant planar integral; if either assumption fails, the predicted QCD alphabets could be incomplete or contain spurious letters.

What would settle it

Compute the symbol of a specific two-loop planar six-point one-mass master integral in the pentagon-triangle topology of figure 4: if the nested square-root letters of eq. (4.22) do not appear, the central prediction fails. Equally decisive would be a three-loop six-point massless computation that either fails to produce the 168 new letters or finds letters outside the predicted set.

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

If this is right

  • The roughly 245-letter six-point one-mass alphabet (29 genuinely six-point letters) is the first concrete target alphabet for the two-loop planar master integrals relevant to vector-boson-plus-three-jets production at the LHC.
  • Nested square-root letters appear in purely polylogarithmic integrals with massless propagators, so existing direct algorithms for determining alphabets from Feynman integrals must be extended; the paper points to momentum twistor variables, where the nested letters simplify.
  • In the massless limit the prediction essentially contains the full finite two-loop six-point massless amplitude alphabet; the letters beyond it (168 by the paper's table, 162 in its abstract) are candidates to appear at higher loops.
  • For five-point two-mass kinematics the prediction reproduces a substantial part of the two-loop alphabet and adds 87 new letters (8 permutation orbits) that are candidates at three-loop order.

Where Pith is reading between the lines

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

  • The mechanism is fully general: any finite alphabet derived from Gr(4,n) cluster data should transfer to (n−k−1)-point (k−1)-mass QCD kinematics. Building the Gr(4,10) alphabet along the same lines would yield the first candidate letters for six-point two-mass and five-point three-mass integrals.
  • The specific gaps in the five-point two-mass comparison — letters built on the r2 root, one of the four non-rationalisable root types, and products of two non-rationalisable roots — may be letters that cancel from the finite two-loop amplitude, mirroring the known pattern that cluster alphabets can contain letters absent from final amplitudes. Analysing the finite function space of that computatio
  • Because the nested square-root letters carry a flip symmetry, the paper's suggestion that they originate from the pentagon-triangle topology of its figure 4 can be settled by directly computing the symbol of that two-loop topology.
  • If the 168 new massless letters appear at three loops, the cluster-algebra route will have outrun direct calculation; if they cancel, it will fit the established phenomenon of cluster alphabets overshooting the letters that actually contribute.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes candidate symbol alphabets for planar 5- and 6-point QCD processes by starting from the conjectural 9-particle N=4 SYM cluster-algebra alphabet of [17], reducing it to (9-k)-point k-mass DCI subalphabets via invariance operators, and then breaking dual conformal invariance to map these to Lorentz-invariant alphabets. The main new results are: (i) a 6-point one-mass alphabet containing 246 (elsewhere 244) letters, including 12 genuinely new letters with nested square roots of the form (4.22); (ii) a massless limit that, after cyclic completion, essentially contains the known 2-loop massless alphabet of [42,43] and yields 168 (abstract says 162) additional letters; and (iii) 5-point two-mass alphabets whose partial overlap with [44] includes 87 new letters. The paper also provides detailed ancillary files and Mathematica code for reproducing the reductions and comparisons.

Significance. If the completeness assumption underlying the DCI-breaking map holds, this work provides a powerful new source of predictions for multi-loop QCD integrals, and the appearance of nested square-root letters in purely polylogarithmic, massless-propagator integrals would be a genuinely novel structural observation. The explicit ancillary data, the reproducible reduction code, and the strong containment of the known 1-loop hexagon and 2-loop massless alphabets are concrete strengths. However, the central step from 'containment' to 'equality' is not proved, and the 5-point two-mass sector demonstrably fails to reproduce some known letter types; hence the new predictions are conditional on an unverified conjecture, and the quantitative claims contain internal inconsistencies that need to be resolved.

major comments (4)
  1. [§1.1] The method rests on the statement that the DCI-broken cluster alphabet is 'certainly contained' in the master-integral alphabet, while equality is 'not excluded'. This is a containment statement, not a proof of equality. All new predictions (Sections 4.4, 5.3, 6.2) require that the cluster alphabet be complete, i.e. equal to the integral alphabet after breaking DCI. No direct integral-level verification of any new letter is provided. Please either supply a proof or a concrete argument for equality in these kinematics, or explicitly qualify every new letter as a candidate under a completeness conjecture and provide at least one independent check (e.g., via Landau equations or a differential-equation calculation) for a representative new letter.
  2. [§6.2, Table 3] The 5-point two-mass comparison explicitly shows that the prediction misses the r2 square-root family and all double-root letters, and the text states this is 'by construction'. This is direct evidence that the DCI-breaking/subalgebra reduction is not complete in a closely related kinematic sector. The manuscript does not explain why the 6-point one-mass and massless predictions should be immune to the same incompleteness. Please add a discussion of the expected validity domain of the method and, if possible, test at least one of the new 6-point massless letters (e.g., β1 or β10) against the known integral results of [42,43].
  3. [Abstract, §1.2, §4.1, §5.3, Table 2, Table 3] The numerical claims are internally inconsistent. The abstract and §1.2 state 162 new massless letters, while Table 2 and §5.3 state 168. The total 6-point one-mass count is 246 in §1.2 but 244 in §4.1 and §5.1. The genuinely new 6-point one-mass letters are described as 8 rational and 9 rationalisable in §1.2, but §4.4 lists 9 rational and 8 rationalisable. In the 5-point two-mass sector, Table 3 implies 11 new orbits (5 rational + 6 Δ5), while §6.2 reports 4 rational + 4 rationalisable = 8 orbits. These discrepancies concern load-bearing claims and must be reconciled before publication.
  4. [§3.1, §7] The input 9-particle SYM alphabet of [17] is itself conjectural, obtained via a stopping criterion for infinite cluster algebras. The paper notes this in §3.1 but does not state in the conclusions that every subsequent prediction inherits this conjecture. If the 9-particle alphabet were incomplete, the predicted QCD alphabets could miss letters even if the DCI-breaking map were exact. Please state this caveat explicitly in Section 7 and discuss any evidence for the completeness of [17] (e.g., agreement with [18] or with Landau-singularity data).
minor comments (6)
  1. [§5.2] The header 'T ranslation' contains an extra space; please fix.
  2. [§4.2] In the ansatz (4.15), the homogeneity in the Mandelstam variables is assumed but not stated; since the letters are scale-invariant, the assignment of polynomial degree should be clarified.
  3. [Eq. (5.7)] The notation '14 sum_{I,J} s_I ϵ_J ∈ W[139,156]' is difficult to read; the sum and the range notation should be typeset more clearly, and the definition of [i,j] referenced.
  4. [§4.4] The paragraph introducing the rational letters says '9 parity even rational letters' and then lists α1...α9, while §1.2 says 8 rational and 9 rationalisable; this mismatch should be corrected consistently.
  5. [§2.4] The displayed matrix in Eq. (2.38) has an obvious formatting artifact (a stray comma after the matrix row); please fix the typesetting.
  6. [§4.1] The word 'straightforwadly' should be 'straightforwardly'.

Circularity Check

0 steps flagged

No significant circularity: the new alphabets are algebraic consequences of a conjectural input alphabet and are checked against independent external benchmarks.

full rationale

The derivation chain starts from the proposed 9-particle N=4 SYM alphabet of [17] and the DCI-breaking procedure of [19], both prior works with overlapping authorship. This is a self-citation, but it is not a circular reduction: the paper explicitly flags the load-bearing completeness/equivalence step as conjectural in Sec. 1.1: 'This will certainly be contained in the alphabet of the master integrals contributing to the amplitude, and one can also not exclude the existence of a good choice of basis where the two coincide.' That is an unproven assumption (a correctness/completeness risk), not an equation reducing a prediction to a fitted input. No parameter of the method is fitted to the target QCD alphabets; the 6-point 1-mass, massless, and 5-point 2-mass letters are obtained by kinematic reduction, shift-invariance constraints, and algebraic reexpression of the input alphabet. The validation against [42,43] and [44] uses externally computed alphabets and reports genuine mismatches (e.g., the missing r2 and double-root families in Sec. 6.2), which would not occur if the result were forced by construction. The small abstract-vs-table discrepancy in the count of new massless letters (162 vs 168) is a numerical/typo issue, not evidence of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no free parameters fitted to data. It relies on four domain assumptions: the completeness of the 9-particle SYM alphabet, the alphabet-preserving nature of DCI breaking, the subsector-containment principle, and the correctness of cluster-algebraic alphabet constructions. There are no invented physical entities; nested square-root letters are mathematical features, not new postulates.

axioms (4)
  • domain assumption The proposed 9-particle N=4 SYM alphabet of [17] is complete for the finite amplitude (starting point of the reduction).
    Section 3.2 calls it a 'candidate finite alphabet'; the paper's predictions inherit its conjectural status.
  • domain assumption Breaking DCI preserves the alphabet: a DCI integral and its LI limit have the same letters under kinematic identification.
    Section 1.1 states this 'should' hold, illustrated by the 4-mass box example, but no general proof is provided.
  • domain assumption The alphabet of a canonical basis contains the alphabets of subsectors (propagator contractions), so lower-point subalphabets are obtained as nullspaces of operators.
    Section 3.3, eq. (3.7) uses this to reduce the 9-point alphabet; it is standard integration-by-parts reasoning but assumed.
  • domain assumption Cluster-algebra/infinite-mutation constructions from [13–18] provide the correct symbol alphabet and radicands for Gr(4,n).
    Section 3.1 reviews this; the paper relies on [17] (self-cited) without re-deriving it.

pith-pipeline@v1.3.0-alltime-deepseek · 39671 in / 14431 out tokens · 127410 ms · 2026-08-02T17:58:46.087924+00:00 · methodology

0 comments
read the original abstract

By breaking dual conformal invariance, we transform cluster-algebraic predictions for the alphabet of 9-point amplitudes in $\mathcal{N}=4$ super Yang-Mills theory to analogous predictions for 5- and 6-point processes in QCD. We start by obtaining, for the first time, candidate letters for 6-point processes with one massive external leg, and discover that they surprisingly also contain nested square roots. We confirm that our results essentially contain the alphabet of all 1-loop integrals with these kinematics, and in their massless limit also the recently computed alphabet of finite, planar 2-loop amplitudes for 6-point massless QCD processes. In the latter case, we additionally find 162 letters that may appear at higher loops. We similarly produce candidate letters for 5-point 2-mass processes, whose comparison with the literature reveals a nontrivial overlap that also includes new letters.

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

83 extracted references · 1 canonical work pages

  1. [1]

    Brink, J.H

    L. Brink, J.H. Schwarz and J. Scherk,Supersymmetric Yang-Mills Theories,Nucl. Phys. B 121(1977) 77

    1977

  2. [2]

    Gliozzi, J

    F. Gliozzi, J. Scherk and D.I. Olive,Supersymmetry, Supergravity Theories and the Dual Spinor Model,Nucl. Phys. B122(1977) 253

    1977

  3. [3]

    Bern, L.J

    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower,One loop n point gauge theory amplitudes, unitarity and collinear limits,Nucl. Phys. B425(1994) 217 [hep-ph/9403226]

    Pith/arXiv arXiv 1994

  4. [4]

    Henn,Multiloop integrals in dimensional regularization made simple,Phys

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

    Pith/arXiv arXiv 2013

  5. [5]

    Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys

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

    1991

  6. [6]

    Remiddi,Differential equations for Feynman graph amplitudes,Nuovo Cim

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

    Pith/arXiv arXiv 1997

  7. [7]

    Gehrmann and E

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

    Pith/arXiv arXiv 2000

  8. [8]

    Chetyrkin and F.V

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

    1981

  9. [9]

    Goncharov, M

    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich,Classical Polylogarithms for Amplitudes and Wilson Loops,Phys. Rev. Lett.105(2010) 151605 [1006.5703]

    Pith/arXiv arXiv 2010

  10. [10]

    Fomin and A

    S. Fomin and A. Zelevinsky,Cluster Algebras I: Foundations,Journal of the American Mathematical Society15(2002) 497 [math/0104151]

    Pith/arXiv arXiv 2002

  11. [11]

    Golden, A.B

    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich,Motivic Amplitudes and Cluster Coordinates,JHEP01(2014) 091 [1305.1617]

    Pith/arXiv arXiv 2014

  12. [12]

    Travaglini et al.,The SAGEX review on scattering amplitudes,J

    G. Travaglini et al.,The SAGEX review on scattering amplitudes,J. Phys. A55(2022) 443001 [2203.13011]

    Pith/arXiv arXiv 2022

  13. [13]

    Drummond, J

    J. Drummond, J. Foster, O. G¨ urdogan and C. Kalousios,Algebraic singularities of scattering amplitudes from tropical geometry,JHEP04(2021) 002 [1912.08217]

    Pith/arXiv arXiv 2021

  14. [14]

    Arkani-Hamed, T

    N. Arkani-Hamed, T. Lam and M. Spradlin,Non-perturbative geometries for planarN= 4 SYM amplitudes,JHEP03(2021) 065 [1912.08222]

    Pith/arXiv arXiv 2021

  15. [15]

    Henke and G

    N. Henke and G. Papathanasiou,How tropical are seven- and eight-particle amplitudes?, JHEP08(2020) 005 [1912.08254]

    Pith/arXiv arXiv 2020

  16. [16]

    Herderschee,Algebraic branch points at all loop orders from positive kinematics and wall crossing,JHEP07(2021) 049 [2102.03611]

    A. Herderschee,Algebraic branch points at all loop orders from positive kinematics and wall crossing,JHEP07(2021) 049 [2102.03611]. – 48 –

    Pith/arXiv arXiv 2021

  17. [17]

    Henke and G

    N. Henke and G. Papathanasiou,Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry,JHEP10(2021) 007 [2106.01392]

    Pith/arXiv arXiv 2021

  18. [18]

    L. Ren, M. Spradlin and A. Volovich,Symbol alphabets from tensor diagrams,JHEP12 (2021) 079 [2106.01405]

    Pith/arXiv arXiv 2021

  19. [19]

    Chicherin, J.M

    D. Chicherin, J.M. Henn and G. Papathanasiou,Cluster algebras for Feynman integrals, Phys. Rev. Lett.126(2021) 091603 [2012.12285]

    Pith/arXiv arXiv 2021

  20. [20]

    Aliaj and G

    R. Aliaj and G. Papathanasiou,An exceptional cluster algebra for Higgs plus jet production, JHEP01(2025) 197 [2408.14544]

    Pith/arXiv arXiv 2025

  21. [21]

    S. He, Z. Li and Q. Yang,Notes on cluster algebras and some all-loop Feynman integrals, JHEP06(2021) 119 [2103.02796]

    Pith/arXiv arXiv 2021

  22. [22]

    S. He, Z. Li, Y. Tang and Q. Yang,Bootstrapping octagons in reduced kinematics from A 2 cluster algebras,JHEP10(2021) 084 [2106.03709]

    Pith/arXiv arXiv 2021

  23. [23]

    S. He, Z. Li and Q. Yang,Truncated cluster algebras and Feynman integrals with algebraic letters,JHEP12(2021) 110 [2106.09314]

    Pith/arXiv arXiv 2021

  24. [24]

    S. He, Z. Li and Q. Yang,Comments on all-loop constraints for scattering amplitudes and Feynman integrals,JHEP01(2022) 073 [2108.07959]

    Pith/arXiv arXiv 2022

  25. [25]

    S. He, Y. Wang, Y. Zhang and P. Zhao,Notes on Worldsheet-Like Variables for Cluster Configuration Spaces,SIGMA19(2023) 045 [2109.13900]

    Pith/arXiv arXiv 2023

  26. [26]

    S. He, Z. Li and Q. Yang,Kinematics, cluster algebras and Feynman integrals,2112.11842

    Pith/arXiv arXiv

  27. [27]

    S. He, Z. Li, R. Ma, Z. Wu, Q. Yang and Y. Zhang,A study of Feynman integrals with uniform transcendental weights and their symbology,JHEP10(2022) 165 [2206.04609]

    Pith/arXiv arXiv 2022

  28. [28]

    S. He, J. Liu, Y. Tang and Q. Yang,The symbology of Feynman integrals from twistor geometries,2207.13482

    Pith/arXiv arXiv

  29. [29]

    Bossinger, J.M

    L. Bossinger, J.M. Drummond and R. Glew,Adjacency for scattering amplitudes from the Gr¨ obner fan,JHEP11(2023) 002 [2212.08931]

    Pith/arXiv arXiv 2023

  30. [30]

    Glew,Holographic correlators and scattering amplitudes: N = 4 SYM and beyond, Ph.D

    R. Glew,Holographic correlators and scattering amplitudes: N = 4 SYM and beyond, Ph.D. thesis

  31. [31]

    Zhao and Y

    P. Zhao and Y. Wang,Cluster alphabets from generalized worldsheets: A geometric approach to finite types,Phys. Rev. D108(2023) 105013

    2023

  32. [32]

    Pokraka, M

    A. Pokraka, M. Spradlin, A. Volovich and H.-C. Weng,Symbol Alphabets in QCD and Flag Cluster Algebras,2506.11895

    arXiv

  33. [33]

    Bossinger, J

    L. Bossinger, J. Drummond, R. Glew, ¨O. G¨ urdo˘ gan and R. Wright,Singularities of massless scattering and cluster algebras,2507.01015

    Pith/arXiv arXiv

  34. [34]

    Bossinger and J.-R

    L. Bossinger and J.-R. Li,Cluster structures on spinor helicity and momentum twistor varieties,2408.14956

    Pith/arXiv arXiv

  35. [35]

    Drummond, J

    J. Drummond, J. Foster and O. G¨ urdo˘ gan,Cluster Adjacency Properties of Scattering Amplitudes inN= 4Supersymmetric Yang-Mills Theory,Phys. Rev. Lett.120(2018) 161601 [1710.10953]

    Pith/arXiv arXiv 2018

  36. [36]

    Gehrmann, J

    T. Gehrmann, J. Henn, P. Jakubˇ c ´ ık, J. Lim, C.C. Mella, N. Syrrakos et al.,Graded transcendental functions: an application to four-point amplitudes with one off-shell leg, JHEP12(2024) 215 [2410.19088]. – 49 –

    Pith/arXiv arXiv 2024

  37. [37]

    Abreu, F

    S. Abreu, F. Febres Cordero, H. Ita, M. Klinkert, B. Page and V. Sotnikov,Leading-color two-loop amplitudes for four partons and a W boson in QCD,JHEP04(2022) 042 [2110.07541]

    Pith/arXiv arXiv 2022

  38. [38]

    Badger, H.B

    S. Badger, H.B. Hartanto, J. Kry´ s and S. Zoia,Two-loop leading colour helicity amplitudes for Wγ+ j production at the LHC,JHEP05(2022) 035 [2201.04075]

    Pith/arXiv arXiv 2022

  39. [39]

    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

  40. [40]

    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

  41. [41]

    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

  42. [42]

    Abreu, P.F

    S. Abreu, P.F. Monni, B. Page and J. Usovitsch,Planar six-point Feynman integrals for four-dimensional gauge theories,JHEP06(2025) 112 [2412.19884]

    Pith/arXiv arXiv 2025

  43. [43]

    J. Henn, A. Matijaˇ si´ c, J. Miczajka, T. Peraro, Y. Xu and Y. Zhang,Complete Function Space for Planar Two-Loop Six-Particle Scattering Amplitudes,Phys. Rev. Lett.135(2025) 031601 [2501.01847]

    Pith/arXiv arXiv 2025

  44. [44]

    Abreu, D

    S. Abreu, D. Chicherin, V. Sotnikov and S. Zoia,Two-loop five-point two-mass planar integrals and double Lagrangian insertions in a Wilson loop,JHEP10(2024) 167 [2408.05201]

    Pith/arXiv arXiv 2024

  45. [45]

    Broadhurst,Summation of an infinite series of ladder diagrams,Physics Letters B307 (1993) 132

    D. Broadhurst,Summation of an infinite series of ladder diagrams,Physics Letters B307 (1993) 132

    1993

  46. [46]

    Drummond, J

    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev,Magic identities for conformal four-point integrals,JHEP01(2007) 064 [hep-th/0607160]

    Pith/arXiv arXiv 2007

  47. [47]

    Alday and J.M

    L.F. Alday and J.M. Maldacena,Gluon scattering amplitudes at strong coupling,JHEP06 (2007) 064 [0705.0303]

    Pith/arXiv arXiv 2007

  48. [48]

    Drummond, G.P

    J.M. Drummond, G.P. Korchemsky and E. Sokatchev,Conformal properties of four-gluon planar amplitudes and Wilson loops,Nucl. Phys. B795(2008) 385 [0707.0243]

    Pith/arXiv arXiv 2008

  49. [49]

    Drummond, J

    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev,Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes,Nucl. Phys. B826(2010) 337 [0712.1223]

    Pith/arXiv arXiv 2010

  50. [50]

    Dirac,Wave equations in conformal space,Annals Math.37(1936) 429

    P.A.M. Dirac,Wave equations in conformal space,Annals Math.37(1936) 429

    1936

  51. [51]

    Usyukina and A.I

    N.I. Usyukina and A.I. Davydychev,An Approach to the evaluation of three and four point ladder diagrams,Phys. Lett. B298(1993) 363

    1993

  52. [52]

    Dlapa, M

    C. Dlapa, M. Helmer, G. Papathanasiou and F. Tellander,Symbol alphabets from the Landau singular locus,JHEP10(2023) 161 [2304.02629]

    Pith/arXiv arXiv 2023

  53. [53]

    Jiang, J

    X. Jiang, J. Liu, X. Xu and L.L. Yang,Symbol letters of Feynman integrals from Gram determinants,2401.07632

    Pith/arXiv arXiv

  54. [54]

    Matijaˇ si´ c,Singularity structure of Feynman integrals with applications to six-particle scattering processes, Ph.D

    A. Matijaˇ si´ c,Singularity structure of Feynman integrals with applications to six-particle scattering processes, Ph.D. thesis, Munich U., 2024. 10.5282/edoc.34154. – 50 –

    work page doi:10.5282/edoc.34154 2024

  55. [55]

    Effortless: Efficient generation of odd letters with multiple roots as leading singularities

    A. Matijaˇ si´ c and J. Miczajka, “Effortless: Efficient generation of odd letters with multiple roots as leading singularities.” https://github.com/antonela-matijasic/Effortless/tree/main, 2025

    2025

  56. [56]

    Correia, M

    M. Correia, M. Giroux and S. Mizera,SOFIA: Singularities of Feynman Integrals Automatized,2503.16601

    arXiv

  57. [57]

    J. Henn, T. Peraro, Y. Xu and Y. Zhang,A first look at the function space for planar two-loop six-particle Feynman integrals,JHEP03(2022) 056 [2112.10605]

    Pith/arXiv arXiv 2022

  58. [58]

    J.M. Henn, A. Matijaˇ si´ c and J. Miczajka,One-loop hexagon integral to higher orders in the dimensional regulator,JHEP01(2023) 096 [2210.13505]

    Pith/arXiv arXiv 2023

  59. [59]

    J.M. Henn, A. Matijaˇ si´ c, J. Miczajka, T. Peraro, Y. Xu and Y. Zhang,A computation of two-loop six-point Feynman integrals in dimensional regularization,JHEP08(2024) 027 [2403.19742]

    Pith/arXiv arXiv 2024

  60. [60]

    9-particle alphabet reduction

    R. Aliaj, G. Dian and G. Papathanasiou, “9-particle alphabet reduction.” https://github.com/GabrieleDian/9-particle-Alphabet-reduction.git

  61. [61]

    Eden, P.V

    R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne,The analytic S-matrix, Cambridge Univ. Press, Cambridge (1966)

    1966

  62. [62]

    Hodges,Eliminating spurious poles from gauge-theoretic amplitudes,JHEP05(2013) 135 [0905.1473]

    A. Hodges,Eliminating spurious poles from gauge-theoretic amplitudes,JHEP05(2013) 135 [0905.1473]

    Pith/arXiv arXiv 2013

  63. [63]

    Arkani-Hamed, J.L

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka,Local Integrals for Planar Scattering Amplitudes,JHEP06(2012) 125 [1012.6032]

    Pith/arXiv arXiv 2012

  64. [64]

    Papathanasiou,The SAGEX review on scattering amplitudes Chapter 5: Analytic bootstraps for scattering amplitudes and beyond,J

    G. Papathanasiou,The SAGEX review on scattering amplitudes Chapter 5: Analytic bootstraps for scattering amplitudes and beyond,J. Phys. A55(2022) 443006 [2203.13016]

    Pith/arXiv arXiv 2022

  65. [65]

    Bourjaily, Y.-H

    J.L. Bourjaily, Y.-H. He, A.J. McLeod, M. Spradlin, C. Vergu, M. Volk et al.,Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail, in Antidifferentiation and the Calculation of Feynman Amplitudes, 3, 2021, DOI [2103.15423]

    Pith/arXiv arXiv 2021

  66. [66]

    Arkani-Hamed, J.L

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (4, 2016), 10.1017/CBO9781316091548, [1212.5605]

    Pith/arXiv arXiv 2016

  67. [67]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang,Scattering Amplitudes,1308.1697

    Pith/arXiv arXiv

  68. [68]

    Britto, F

    R. Britto, F. Cachazo, B. Feng and E. Witten,Direct proof of tree-level recursion relation in Yang-Mills theory,Phys. Rev. Lett.94(2005) 181602 [hep-th/0501052]

    Pith/arXiv arXiv 2005

  69. [69]

    Drummond, J.M

    J.M. Drummond, J.M. Henn and J. Trnka,New differential equations for on-shell loop integrals,JHEP04(2011) 083 [1010.3679]

    Pith/arXiv arXiv 2011

  70. [70]

    Speyer and L

    D. Speyer and L. Williams,The Tropical Totally Positive Grassmannian,Journal of Algebraic Combinatorics22(2005) 189 [math/0312297]

    Pith/arXiv arXiv 2005

  71. [71]

    Canakci and R

    I. Canakci and R. Schiffler,Cluster algebras and continued fractions,Compositio Mathematica154(2018) 565 [1608.06568]

    Pith/arXiv arXiv 2018

  72. [72]

    Reading,A combinatorial approach to scattering diagrams,1806.05094

    N. Reading,A combinatorial approach to scattering diagrams,1806.05094

    Pith/arXiv arXiv

  73. [73]

    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]. – 51 –

    Pith/arXiv arXiv 2019

  74. [74]

    Binoth, J.P

    T. Binoth, J.P. Guillet and G. Heinrich,Reduction formalism for dimensionally regulated one loop N point integrals,Nucl. Phys. B572(2000) 361 [hep-ph/9911342]

    Pith/arXiv arXiv 2000

  75. [75]

    Abreu, D

    S. Abreu, D. Chicherin, H. Ita, B. Page, V. Sotnikov, W. Tschernow et al.,All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering,2306.15431

    Pith/arXiv arXiv

  76. [76]

    Mitev and Y

    V. Mitev and Y. Zhang,SymBuild: a package for the computation of integrable symbols in scattering amplitudes,1809.05101

    Pith/arXiv arXiv

  77. [77]

    Besier, P

    M. Besier, P. Wasser and S. Weinzierl,RationalizeRoots: Software Package for the Rationalization of Square Roots,Comput. Phys. Commun.253(2020) 107197 [1910.13251]

    Pith/arXiv arXiv 2020

  78. [78]

    Bitoun, C

    T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer,Feynman integral relations from parametric annihilators,Lett. Math. Phys.109(2019) 497 [1712.09215]

    Pith/arXiv arXiv 2019

  79. [79]

    Fevola, S

    C. Fevola, S. Mizera and S. Telen,Landau Singularities Revisited: Computational Algebraic Geometry for Feynman Integrals,Phys. Rev. Lett.132(2024) 101601 [2311.14669]

    Pith/arXiv arXiv 2024

  80. [80]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Trnka,The Amplituhedron,JHEP10(2014) 030 [1312.2007]

    Pith/arXiv arXiv 2014

Showing first 80 references.