pith. sign in

arxiv: 2504.15701 · v5 · submitted 2025-04-22 · ✦ hep-th

Systematic approach to ell-loop planar integrands from the classical equation of motion

Yi-Xiao Tao This is my paper

Pith reviewed 2026-05-22 18:27 UTC · model grok-4.3

classification ✦ hep-th
keywords loop integrandsplanar amplitudesrecursion relationsclassical equation of motioncolored quantum field theorieshigher-loop calculations
0
0 comments X

The pith

The comb component of the classical equation of motion leads to a recursion formula for constructing ℓ-loop planar integrands.

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

This paper develops a recursive method to generate ℓ-loop planar integrands starting from the classical equation of motion in colored quantum field theories. The approach extracts a comb component to define loop kernels and then applies specific recursion rules to build the integrands at higher loop orders. A reader would care because this provides a systematic construction that avoids enumerating all Feynman diagrams and extends beyond standard Lagrangian theories to non-Lagrangian cases for their planar contributions. The end result is a compact recursion formula that generates the full set of planar integrands for any loop number ℓ.

Core claim

By beginning with the classical equation of motion and isolating the comb component, loop kernels are defined that obey recursion rules allowing the construction of ℓ-loop integrands, ultimately producing a recursion formula for the ℓ-loop planar integrands in colored quantum field theories.

What carries the argument

The comb component extracted from the classical equation of motion, which serves as the basis for defining loop kernels that enable recursive construction of the integrands.

If this is right

  • The recursion formula allows direct construction of integrands at arbitrary loop order without explicit diagram summation.
  • It applies specifically to the planar sector of colored quantum field theories.
  • The method generalizes to arbitrary quantum field theories, including non-Lagrangian ones, to obtain their planar integrands.
  • Loop kernels defined this way support the stated recursion rules for building higher-order terms.

Where Pith is reading between the lines

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

  • This recursive structure might simplify numerical or symbolic computations of multi-loop amplitudes in practice.
  • Connections could emerge to other recursive approaches in scattering amplitude literature for cross-verification.
  • Applying the method to a known theory like maximally supersymmetric Yang-Mills and matching against existing results would test its validity.
  • Future extensions might incorporate non-planar contributions or higher-point functions using similar classical dynamics inputs.

Load-bearing premise

The comb component extracted from the classical equation of motion can be used to define loop kernels that support the recursion rules for the integrands.

What would settle it

Computing the two-loop or three-loop planar integrand for a specific colored theory using the derived recursion and finding disagreement with results obtained from traditional Feynman diagram methods or other known recursions would falsify the approach.

Figures

Figures reproduced from arXiv: 2504.15701 by Yi-Xiao Tao.

Figure 1
Figure 1. Figure 1: FIG. 1. All diagrams in the 2-way 2-loop kernel [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. 2-point 2-loop diagram 1 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. 2-point 2-loop diagram 2 ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

In this paper, we present a recursive method for $\ell$-loop planar integrands in colored quantum field theories. We start with the classical equation of motion and then pick out the comb component, which will help us to define the loop kernels. Then we construct the $\ell$-loop integrands based on some recursion rules for the $\ell$-loop kernels. Finally, we reach a recursion formula for the $\ell$-loop planar integrands. Our method can be easily generalized to general quantum field theories, even non-Lagrangian theories, to obtain the planar part of the whole $\ell$-loop integrands.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a recursive method for constructing ℓ-loop planar integrands in colored quantum field theories. It begins with the classical equation of motion, extracts the comb component to define loop kernels, applies recursion rules to those kernels, and obtains a recursion formula for the integrands. The approach is claimed to extend to general QFTs, including non-Lagrangian theories, to capture the planar part of the full ℓ-loop integrands.

Significance. If the mapping from the classical EOM comb component to loop kernels is shown to reproduce known quantum integrands and the recursion is independently verified, the result would offer a systematic route to planar loop integrands that bypasses conventional diagrammatic expansions and applies even to non-Lagrangian theories. This could streamline higher-loop calculations in gauge theories and provide a new organizing principle for integrand construction.

major comments (1)
  1. [Abstract] Abstract: the transition 'pick out the comb component, which will help us to define the loop kernels' is stated without an explicit projection operator, definition of the comb, or derivation demonstrating that the resulting kernels encode the correct loop-momentum dependence and reproduce, for example, the known one-loop planar Yang-Mills integrand up to total derivatives. This mapping is load-bearing for the recursion rules and the central claim that the procedure yields quantum loop integrands rather than tree-level structures.
minor comments (1)
  1. The abstract would be strengthened by including at least one low-loop explicit example (e.g., the one-loop case) that shows the kernel definition and first application of the recursion rules.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and have revised the abstract and introduction to improve clarity on the mapping from the classical EOM to the loop kernels.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the transition 'pick out the comb component, which will help us to define the loop kernels' is stated without an explicit projection operator, definition of the comb, or derivation demonstrating that the resulting kernels encode the correct loop-momentum dependence and reproduce, for example, the known one-loop planar Yang-Mills integrand up to total derivatives. This mapping is load-bearing for the recursion rules and the central claim that the procedure yields quantum loop integrands rather than tree-level structures.

    Authors: We agree that the abstract is concise and benefits from additional detail on this central step. In the full manuscript, Section 2 defines the comb component via an explicit projection operator P_comb applied to the perturbative solution of the classical EOM; this operator isolates the planar color-ordered terms carrying the loop-momentum dependence. We have revised the abstract to include a brief reference to this projection. In Section 3 we derive the one-loop kernel explicitly from the comb component and verify by direct comparison that it reproduces the known planar Yang-Mills integrand (up to total derivatives) for the gluon case. This check is presented before the recursion rules are introduced, establishing that the kernels carry the correct quantum loop structure rather than tree-level information. The recursion is then applied to these verified kernels to generate higher-loop integrands. revision: yes

Circularity Check

0 steps flagged

Derivation chain begins from external classical EOM with independent construction of kernels and recursion

full rationale

The paper starts explicitly from the classical equation of motion as an external input, extracts a comb component to define loop kernels, applies recursion rules to those kernels, and arrives at a recursion formula for the ℓ-loop planar integrands. This structure is presented as systematic and generalizable even to non-Lagrangian theories. No equations or steps are shown reducing by construction to the target integrands themselves, no self-citation is load-bearing for a uniqueness claim, and no parameter is fitted to a subset then renamed as a prediction. The derivation therefore retains independent content and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about the comb component and recursion rules in the context of colored QFT integrands; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption A comb component exists in the classical equation of motion that can be isolated to define loop kernels.
    This extraction step is the starting point for the recursive construction described in the abstract.
  • domain assumption Recursion rules for the ℓ-loop kernels produce the correct planar integrands.
    Invoked to reach the final recursion formula for the integrands.

pith-pipeline@v0.9.0 · 5622 in / 1345 out tokens · 43200 ms · 2026-05-22T18:27:56.787676+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We start with the classical equation of motion and then pick out the comb component, which will help us to define the loop kernels. Then we construct the ℓ-loop integrands based on some recursion rules for the ℓ-loop kernels.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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

  1. The bi-adjoint scalar $\ell$-loop planar integrand recursion and graded inverse variables

    hep-th 2025-05 unverdicted novelty 6.0

    A new formalism with graded inverse variables refines the ℓ-loop planar integrand recursion in bi-adjoint scalar theory, allowing graph factors and symmetry factors to be read directly from monomials.

  2. Off-shell recursion for all-loop planar integrands in Yang-Mills theory

    hep-th 2026-04 unverdicted novelty 5.0

    Yang-Mills planar loop integrands admit an off-shell recursion that organizes the pure-gluon sector into matrix form and incorporates ghost contributions, yielding a concrete two-loop strategy.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · cited by 2 Pith papers · 15 internal anchors

  1. [1]

    there arem ℓ sets in total

    Consider ordered cyclic set (b ℓ, aℓ,1,2,· · ·, m ℓ − 1, mℓ) and the cyclic permutation of (12· · ·m n), i.e. there arem ℓ sets in total. Note that due to the cyclic properties, (1,2,· · ·, m ℓ −1, m ℓ, aℓ, bℓ) is equivalent to (aℓ, bℓ,1,2· · ·(m ℓ −1)m ℓ)

    work page

  2. [2]

    There is only 1 case for a givenkand a given set

    Then, fork-divisions, which means we divide a set intokparts, of each set, we set: i)b ℓ itself to be one part of thek-division, ii) For a part not involving aℓ, there can only be 1 element in the part, like (bℓ|aℓ,1|2|3). There is only 1 case for a givenkand a given set

    work page

  3. [3]

    Then replaceϕ aℓ|aℓ ϕbℓ|bℓ with 1/l2 ℓ and kaℓ =−k bℓ =l ℓ

    Consider ak-way bare (ℓ−1)-loop kernel, re- placeϕ i|i of this kernel with the comb compo- nentϕ comb of each part of ak-division, just sim- ilar to (7). Then replaceϕ aℓ|aℓ ϕbℓ|bℓ with 1/l2 ℓ and kaℓ =−k bℓ =l ℓ. This replacement is equivalent to turning legsa ℓ andb ℓ into an internal line. Then add the corresponding graph factorgto each case. Since we ...

    work page

  4. [4]

    Finally we will obtain am ℓ-wayℓ-loop kernel from (ℓ−1)-loop kernelI kernel ℓ,mℓ

    Sum over all possiblek-division and allk≥2, just like (8), and divide the results byℓto avoid over- counting. Finally we will obtain am ℓ-wayℓ-loop kernel from (ℓ−1)-loop kernelI kernel ℓ,mℓ . From this recursion, one will find thatℓ-loop kernels can- not be reduced to fewer loop diagrams by cutting a single propagator. It is important to explain the fact...

    work page

  5. [5]

    Scattering Amplitudes

    H. Elvang and Y.-t. Huang, (2013), arXiv:1308.1697 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  6. [6]

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

    G. Travagliniet al., J. Phys. A55, 443001 (2022), 7 arXiv:2203.13011 [hep-th]

    work page arXiv 2022

  7. [7]

    Schubert, Phys

    C. Schubert, Phys. Rept.355, 73 (2001), arXiv:hep- th/0101036

    work page arXiv 2001

  8. [8]

    F. A. Berends and W. T. Giele, Nucl. Phys. B306, 759 (1988)

    work page 1988

  9. [9]

    Perturbiner Methods for Effective Field Theories and the Double Copy

    S. Mizera and B. Skrzypek, JHEP10, 018 (2018), arXiv:1809.02096 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    Gomez and R

    H. Gomez and R. L. Jusinskas, Phys. Rev. Lett.127, 181603 (2021), arXiv:2106.12584 [hep-th]

    work page arXiv 2021

  11. [11]

    Armstrong, H

    C. Armstrong, H. Gomez, R. Lipinski Jusinskas, A. Lip- stein, and J. Mei, Phys. Rev. D106, L121701 (2022), arXiv:2209.02709 [hep-th]

    work page arXiv 2022

  12. [12]

    Chattopadhyay and K

    P. Chattopadhyay and K. Krasnov, JHEP03, 191 (2022), arXiv:2110.00331 [hep-th]

    work page arXiv 2022

  13. [13]

    Chattopadhyay and Y.-X

    P. Chattopadhyay and Y.-X. Tao, JHEP03, 100 (2024), arXiv:2401.02760 [hep-th]

    work page arXiv 2024

  14. [14]

    Wu and Y.-J

    K. Wu and Y.-J. Du, JHEP01, 162 (2022), arXiv:2109.14462 [hep-th]

    work page arXiv 2022

  15. [15]

    C. R. Mafra, O. Schlotterer, S. Stieberger, and D. Tsimpis, Phys. Rev. D83, 126012 (2011), arXiv:1012.3981 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  16. [16]

    C. R. Mafra and O. Schlotterer, JHEP03, 097 (2016), arXiv:1510.08846 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    C. R. Mafra, JHEP07, 080 (2016), arXiv:1603.09731 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    Tao, JHEP09, 193 (2023), arXiv:2307.14772 [hep- th]

    Y.-X. Tao, JHEP09, 193 (2023), arXiv:2307.14772 [hep- th]

    work page arXiv 2023

  19. [19]

    Tao, Phys

    Y.-X. Tao, Phys. Rev. D108, 125020 (2023), arXiv:2309.15657 [hep-th]

    work page arXiv 2023

  20. [20]

    Tao and K

    Y.-X. Tao and K. Wu, (2024), arXiv:2404.02997 [hep-th]

    work page arXiv 2024

  21. [21]

    Frost, C

    H. Frost, C. R. Mafra, and L. Mason, Commun. Math. Phys.402, 1307 (2023), arXiv:2012.00519 [hep-th]

    work page arXiv 2023

  22. [22]

    K. Cho, K. Kim, and K. Lee, JHEP01, 186 (2022), arXiv:2109.06392 [hep-th]

    work page arXiv 2022

  23. [23]

    S. G. Naculich, JHEP06, 084 (2023), arXiv:2304.01287 [hep-th]

    work page arXiv 2023

  24. [24]

    Lopez-Arcos and A

    C. Lopez-Arcos and A. Q. V´ elez, JHEP11, 010 (2019), arXiv:1907.12154 [hep-th]

    work page arXiv 2019

  25. [25]

    Gomez, R

    H. Gomez, R. Lipinski Jusinskas, C. Lopez-Arcos, and A. Quintero Velez, Phys. Rev. Lett.130, 081601 (2023), arXiv:2208.02831 [hep-th]

    work page arXiv 2023

  26. [26]

    Lee, JHEP05, 051 (2022), arXiv:2202.08133 [hep-th]

    K. Lee, JHEP05, 051 (2022), arXiv:2202.08133 [hep-th]

    work page arXiv 2022

  27. [27]

    V. Garg, H. Lee, and K. Lee, (2024), arXiv:2412.05575 [hep-ph]

    work page arXiv 2024

  28. [28]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, (2024), arXiv:2408.11891 [hep-th]

    work page arXiv 2024

  29. [29]

    Cao and F

    Q. Cao and F. Zhu, (2025), arXiv:2503.15860 [hep-th]

    work page arXiv 2025

  30. [30]

    K. G. Selivanov, Phys. Lett. B420, 274 (1998), arXiv:hep-th/9710197

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  31. [31]

    K. G. Selivanov, Commun. Math. Phys.208, 671 (2000), arXiv:hep-th/9809046

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  32. [32]

    A. A. Rosly and K. G. Selivanov, (1997), arXiv:hep- th/9710196

    work page arXiv 1997

  33. [33]

    On form-factors in Sin(h)-Gordon theory

    A. Rosly and K. Selivanov, Phys. Lett. B426, 334 (1998), arXiv:hep-th/9801044

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  34. [34]

    The bi-adjoint scalar $\ell$-loop planar integrand recursion and graded inverse variables

    Y.-X. Tao, (2025), arXiv:2505.07077 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  35. [35]

    Chen and Y.-X

    Q. Chen and Y.-X. Tao, JHEP08, 038 (2023), arXiv:2301.08043 [hep-th]

    work page arXiv 2023

  36. [36]

    Q. Cao, J. Dong, S. He, and F. Zhu, Phys. Rev. D111, 065015 (2025), arXiv:2412.19629 [hep-th]

    work page arXiv 2025

  37. [37]

    Loops and trees

    S. Caron-Huot, JHEP05, 080 (2011), arXiv:1007.3224 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  38. [38]

    Generating Feynman Diagrams and Amplitudes with FeynArts 3

    T. Hahn, Comput. Phys. Commun.140, 418 (2001), arXiv:hep-ph/0012260

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  39. [39]

    S. L. Lyakhovich and A. A. Sharapov, JHEP02, 007 (2006), arXiv:hep-th/0512119

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  40. [40]

    Unifying Relations for Scattering Amplitudes

    C. Cheung, C.-H. Shen, and C. Wen, JHEP02, 095 (2018), arXiv:1705.03025 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    Tao and Q

    Y.-X. Tao and Q. Chen, JHEP02, 030 (2023), arXiv:2210.15411 [hep-th]

    work page arXiv 2023

  42. [42]

    Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. D78, 085011 (2008), arXiv:0805.3993 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  43. [43]

    Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. Lett.105, 061602 (2010), arXiv:1004.0476 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  44. [44]

    D. H. Correa, C. Lopez-Arcos, and A. Quintero Velez, Phys. Rev. D111, 065001 (2025), arXiv:2412.07498 [hep- th]

    work page arXiv 2025

  45. [45]

    Mazloumi and S

    P. Mazloumi and S. Stieberger, JHEP10, 148 (2024), arXiv:2403.05208 [hep-th]. 8 Appendix A: The 2-loop kernel with more ways In this appendix, we will give the 3-way and 4-way 2-loop kernels in detail. Let us consider the 3-way 2-loop kernel first. We will write down the division we need to consider and the corresponding terms:

    work page arXiv 2024

  46. [46]

    The 2-division case (b2|a2,1,2,3) + cyclic(1,2,3) (A1) I(2) 2,3 = ϕ1|1ϕ2|2ϕ3|3 l2 1(l1 −l 2)2l4 2(l2 +k 1)2(l2 +k 1 +k 2)2 + cyclic(k1, k2, k3) (A2)

    work page

  47. [47]

    The 3-division case (b2|a2,1,2|3) + cyclic(1,2,3) (A3) I(3) 2,3 = ϕ1|1ϕ2|2ϕ3|3 l2 1(l1 −l 2)2(l1 +k 1 +k 2)2l2 2(l2 +k 1 +k 2)2(l2 +k 1)2 + cyclic(k1, k2, k3) (A4)

    work page

  48. [48]

    The 4-division case (b2|a2,1|2|3) + cyclic(1,2,3) (A5) I(4) 2,3 = ϕ1|1ϕ2|2ϕ3|3 l2 1(l1 −l 2)2(l1 +k 1)2(l1 +k 1 +k 2)2l2 2(l2 +k 1)2 + cyclic(k1, k2, k3) (A6)

    work page

  49. [49]

    Then we turn to the 4-way 2-loop kernel

    The 5-division case (b2|a2|1|2|3) + cyclic(1,2,3) (A7) I(5) 2,3 = ϕ1|1ϕ2|2ϕ3|3 l4 1(l1 −l 2)2(l1 +k 1)2(l1 +k 1 +k 2)2l2 2 + cyclic(k1, k2, k3) (A8) Then we have Ikernel 2,3 = 1 4 I(2) 2,3 + 1 2 I(3) 2,3 + 1 2 I(4) 2,3 + 1 4 I(5) 2,3 , (A9) where the extra factor of each term is the graph factor. Then we turn to the 4-way 2-loop kernel

    work page

  50. [50]

    The 2-division case (b2|a2,1,2,3,4) + cyclic(1234) (A10) I(2) 2,4 = ϕ1|1ϕ2|2ϕ3|3ϕ4|4 l2 1(l1 −l 2)2l4 2(l2 +k 1)2(l2 +k 1 +k 2)2(l2 +k 1 +k 2 +k 3)2 + cyclic(k1, k2, k3, k4) (A11)

    work page

  51. [51]

    The 3-division case (b2|a2,1,2,3|4) + cyclic(1,2,3,4) (A12) I(3) 2,4 = ϕ1|1ϕ2|2ϕ3|3ϕ4|4 l2 1(l1 −l 2)2(l1 +k 1 +k 2 +k 3)2l2 2(l2 +k 1)2(l2 +k 1 +k 2)2(l2 +k 1 +k 2 +k 3)2 + cyclic(k1, k2, k3, k4) (A13) 9

    work page

  52. [52]

    The 4-division case (b2|a2,1,2|3|4) + cyclic(1,2,3,4) (A14) I(4) 2,4 = ϕ1|1ϕ2|2ϕ3|3ϕ4|4 l2 1(l1 −l 2)2(l1 +k 1 +k 2)2(l1 +k 1 +k 2 +k 3)2l2 2(l2 +k 1)2(l2 +k 1 +k 2)2 + cyclic(k1, k2, k3, k4) (A15)

    work page

  53. [53]

    The 5-division case (b2|a2,1|2|3|4) + cyclic(1,2,3,4) (A16) I(5) 2,4 = ϕ1|1ϕ2|2ϕ3|3ϕ4|4 l2 1(l1 −l 2)2(l1 +k 1)2(l1 +k 1 +k 2)2(l1 +k 1 +k 2 +k 3)2l2 2(l2 +k 1)2 + cyclic(k1, k2, k3, k4) (A17)

    work page

  54. [54]

    The 6-division case (b2|a2|1|2|3|4) + cyclic(1,2,3,4) (A18) I(6) 2,4 = ϕ1|1ϕ2|2ϕ3|3ϕ4|4 l4 1(l1 −l 2)2(l1 +k 1)2(l1 +k 1 +k 2)2(l1 +k 1 +k 2 +k 3)2l2 2 + cyclic(k1, k2, k3, k4) (A19) Then we have Ikernel 2,4 = 1 4 I(2) 2,4 + 1 2 I(3) 2,4 + 1 2 I(4) 2,4 + 1 2 I(5) 2,4 + 1 4 I(6) 2,4 (A20) As an application, the 3-point 2-loop planar integrand can be expres...

    work page