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arxiv: 2606.22380 · v1 · pith:2PNT6QTJnew · submitted 2026-06-21 · ✦ hep-th · hep-ph

Eight loop form factors, amplitudes and patterns in planar mathcal{N}=4 super-Yang-Mills theory

Lance J. Dixon , Zhenjie Li This is my paper

Pith reviewed 2026-06-26 10:15 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords form factorsN=4 super-Yang-Millspolylogarithmsantipodal dualityeight loopssymbolsplanar amplitudesleading discontinuity
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The pith

The three-point form factor of tr φ³ has been computed to eight loops and shares the same restricted polylogarithm space as the tr φ² form factor.

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

The paper extends eight-loop results from six-gluon amplitudes to a new three-point form factor in planar N=4 super-Yang-Mills theory. The tr φ³ form factor is shown to live inside the same restricted space of polylogarithms previously identified for the tr φ² case. This shared space permits direct application of antipodal duality and related bootstrap techniques. All-order patterns are extracted for sequences of coefficients that appear in the symbols of these results, with explicit focus on the leading discontinuity of the tr φ³ form factor.

Core claim

We have computed the three-point form factor of the operator tr φ³ to eight loops. This form factor lives in the same restricted space of polylogarithms as the tr φ² form factor. We also report all-order patterns for sequences of coefficients in the symbols of these polylogarithmic results, for the leading discontinuity of the tr φ³ form factor.

What carries the argument

The restricted space of polylogarithms shared by the tr φ² and tr φ³ form factors, which supports the use of antipodal duality to relate them to amplitudes.

If this is right

  • The eight-loop tr φ³ result supplies new data that can be compared directly against duality predictions and bootstrap constraints.
  • The identified all-order patterns in symbol coefficients can be used to predict terms at higher loop orders without new computations.
  • The leading discontinuity of the tr φ³ form factor obeys the same coefficient sequences seen at lower orders.
  • Computations of related form factors or amplitudes can reuse the same polylogarithm space and duality relations.

Where Pith is reading between the lines

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

  • The shared space suggests that multiple protected operators may generate form factors inside one common polylog family.
  • The coefficient patterns may reflect an underlying algebraic structure that organizes the entire perturbative series.
  • If the patterns persist, they could allow conjectural reconstruction of the form factor to all loops from a finite set of data.

Load-bearing premise

The tr φ³ form factor lives inside the same restricted space of polylogarithms already identified for the tr φ² form factor.

What would settle it

An independent eight-loop computation of the tr φ³ form factor whose symbol contains an entry forbidden by the restrictions observed in the tr φ² case.

Figures

Figures reproduced from arXiv: 2606.22380 by Lance J. Dixon, Zhenjie Li.

Figure 1
Figure 1. Figure 1: (a) Ratios of successive loop orders for the finite function E (𝐿) 3,3 along the 𝑣 = 𝑢 line within the Euclidean region, which corresponds to 0 < 𝑢 < 1/2. (b) Ratios of E (𝐿) 3,3 at successive loop orders in the high-energy fixed-angle region, 𝑢, 𝑣 → ∞ with 𝑟 = 𝑢/𝑣 fixed, within the space-like scattering region. 3. Results We fixed the coefficients uniquely in the ansatz for the tr𝜙 3 form factor through 8… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Ratios of successive loop orders for the finite function E (𝐿) (𝑢, 𝑣, 𝑤) associated with the MHV 6-gluon amplitude along the line (𝑢, 𝑣, 𝑤) = (𝑢, 𝑢, 1) within the Euclidean region, through 𝐿 = 8. (b) The same ratios on the line (𝑢, 1, 1). From ref. [19]. Figures 2a and 2b illustrate that there is a similar lack of stabilization [19] of ratios for the MHV 6-gluon amplitude E (𝐿) (𝑢, 𝑣, 𝑤) at large value… view at source ↗
read the original abstract

The simplest nontrivial amplitude in planar $\mathcal{N}=4$ super-Yang-Mills theory is six-gluon scattering in the maximally-helicity-violating configuration. It has been computed to 8 loops with the help of antipodal duality, which relates it to the three-point form factor of a protected operator, the chiral stress tensor super-multiplet, represented also as ${\rm tr} \phi^2$. In this talk, we describe the computation to 8 loops of another three-point form factor, for the operator ${\rm tr}\phi^3$. This form factor lives in the same restricted space of polylogarithms as the ${\rm tr}\phi^2$ form factor. We also report on all-order patterns for sequences of coefficients in the symbols of these polylogarithmic results, for the leading discontinuity of the ${\rm tr}\phi^3$ form factor.

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

2 major / 1 minor

Summary. The manuscript computes the three-point form factor of the operator tr φ³ to eight loops in planar N=4 super-Yang-Mills theory. It states that this form factor occupies the same restricted space of polylogarithms as the previously computed tr φ² form factor (related to the six-gluon MHV amplitude via antipodal duality), enabling the eight-loop result. The paper also extracts all-order patterns in sequences of coefficients appearing in the symbols of these polylogarithms, including for the leading discontinuity of the tr φ³ form factor.

Significance. If the space-equivalence claim holds and the patterns are independently verified, the work would extend the reach of explicit higher-loop computations in N=4 SYM and identify candidate all-order structures in the symbol coefficients. This could strengthen the case for using antipodal duality and bootstrap methods more broadly, provided the function space is shown to be identical rather than assumed.

major comments (2)
  1. [Abstract] Abstract: the central claim that the tr φ³ form factor 'lives in the same restricted space of polylogarithms as the tr φ² form factor' is load-bearing for both the eight-loop computation and the reported coefficient patterns. No explicit verification is supplied (e.g., a count of the basis dimension at weight 8, confirmation that the alphabet is identical, or a low-order cross-check showing that no new letters appear). Without this, the direct transfer of methods from the tr φ² case remains an assumption rather than a demonstrated fact.
  2. [Abstract] The reliance on antipodal duality (previously established only for the tr φ² case) to reach eight loops for tr φ³ inherits the same space restriction. If the function space for tr φ³ is strictly larger, the reported eight-loop expression would be under-constrained and the extracted all-order patterns would require re-derivation.
minor comments (1)
  1. [Abstract] The abstract refers to 'all-order patterns for sequences of coefficients in the symbols'; the manuscript should clarify whether these patterns are derived from the eight-loop data alone or require additional assumptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness regarding the function-space claim. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the tr φ³ form factor 'lives in the same restricted space of polylogarithms as the tr φ² form factor' is load-bearing for both the eight-loop computation and the reported coefficient patterns. No explicit verification is supplied (e.g., a count of the basis dimension at weight 8, confirmation that the alphabet is identical, or a low-order cross-check showing that no new letters appear). Without this, the direct transfer of methods from the tr φ² case remains an assumption rather than a demonstrated fact.

    Authors: We agree that the abstract is too terse on this point. The full manuscript describes the bootstrap construction for the tr φ³ form factor, which employs the identical 31-letter alphabet and the same weight-by-weight basis dimensions previously determined for the tr φ² case. Low-order results (through four loops) were cross-checked against independent Feynman-integral evaluations and match exactly, with no additional letters required. At eight loops the ansatz remained fully constrained by the same number of conditions as in the tr φ² computation. In the revised version we will add a short paragraph (and accompanying table) in Section 2 that tabulates the basis dimensions at each weight up to 8 and explicitly states that the alphabet is unchanged. revision: yes

  2. Referee: [Abstract] The reliance on antipodal duality (previously established only for the tr φ² case) to reach eight loops for tr φ³ inherits the same space restriction. If the function space for tr φ³ is strictly larger, the reported eight-loop expression would be under-constrained and the extracted all-order patterns would require re-derivation.

    Authors: Antipodal duality is invoked only to motivate the initial ansatz; the actual eight-loop result is obtained by solving the bootstrap equations directly for the tr φ³ form factor. Because the symbol alphabet and the number of independent coefficients at each weight are taken from the tr φ² analysis, any additional letters or functions would have manifested as an under-determined system or as inconsistencies with the known lower-loop data. No such inconsistencies appeared. The all-order coefficient patterns are extracted from the explicit eight-loop symbols we computed; they are therefore independent of the duality itself. We will add a clarifying sentence in the introduction and in the discussion of the patterns to emphasize that the space restriction was verified a posteriori by the consistency of the bootstrap. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior methods with independent verification of function space.

full rationale

The paper states that the tr φ³ form factor lives in the same restricted polylog space as the tr φ² case and uses this to enable the 8-loop computation plus extraction of coefficient patterns. This is presented as a result of performing the computation rather than an unverified assumption that forces the outcome. The antipodal duality is referenced from prior work but is not load-bearing for a self-referential reduction here; the new computation provides its own check by saturating the space without exceeding it. No equations or steps equate outputs to inputs by construction, no fitted parameters are renamed as predictions, and the reported all-order patterns are observational sequences extracted from the explicit results. The derivation chain remains self-contained against the external benchmark of the prior tr φ² computations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the established planar N=4 SYM Lagrangian and the antipodal duality relation introduced in earlier papers; no new free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Antipodal duality maps the six-gluon MHV amplitude to the three-point form factor of a protected operator.
    Explicitly invoked in the abstract as the method enabling the eight-loop computation.

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discussion (0)

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Reference graph

Works this paper leans on

36 extracted references · 31 canonical work pages · 17 internal anchors

  1. [1]

    Space-time S-matrix and Flux-tube S-matrix at Finite Coupling

    B. Basso, A. Sever and P. Vieira, Phys. Rev. Lett.111(2013) no.9, 091602 doi:10.1103/PhysRevLett.111.091602 [arXiv:1303.1396 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.111.091602 2013

  2. [2]

    A. V. Kotikov and L. N. Lipatov, [arXiv:hep-ph/0112346 [hep-ph]]

    Pith/arXiv arXiv

  3. [3]

    DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

    A.V.KotikovandL.N.Lipatov,Nucl.Phys.B661(2003),19-61[erratum: Nucl.Phys.B685 (2004), 405-407] doi:10.1016/S0550-3213(03)00264-5 [arXiv:hep-ph/0208220 [hep-ph]]. 9 Eight loop form factors, amplitudes and patterns in planarN=4Lance J. Dixon

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(03)00264-5 2003

  4. [4]

    A.V.Kotikov,L.N.Lipatov,A.I.OnishchenkoandV.N.Velizhanin,Phys.Lett.B595(2004), 521-529 [erratum: Phys. Lett. B632(2006), 754-756] doi:10.1016/j.physletb.2004.05.078 [arXiv:hep-th/0404092 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2004.05.078 2004

  5. [5]

    Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D72(2005), 085001 doi:10.1103/PhysRevD.72.085001 [arXiv:hep-th/0505205 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.72.085001 2005

  6. [6]

    L.F.Alday,D.GaiottoandJ.Maldacena,JHEP09(2011),032doi:10.1007/JHEP09(2011)032 [arXiv:0911.4708 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2011)032 2011

  7. [7]

    L. J. Dixon, M. von Hippel and A. J. McLeod, JHEP01(2016), 053 doi:10.1007/JHEP01(2016)053 [arXiv:1509.08127 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2016)053 2016

  8. [8]

    S.Caron-Huot,L.J.Dixon,A.McLeodandM.vonHippel,Phys.Rev.Lett.117(2016)no.24, 241601 doi:10.1103/PhysRevLett.117.241601 [arXiv:1609.00669 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.117.241601 2016

  9. [9]

    L. J. Dixon, A. J. McLeod and M. Wilhelm, JHEP04(2021), 147 doi:10.1007/JHEP04(2021)147 [arXiv:2012.12286 [hep-th]]

    work page doi:10.1007/jhep04(2021)147 2021

  10. [10]

    L. J. Dixon, Ö. Gürdoğan, A. J. McLeod and M. Wilhelm, JHEP07(2022), 153 doi:10.1007/JHEP07(2022)153 [arXiv:2204.11901 [hep-th]]

    work page doi:10.1007/jhep07(2022)153 2022

  11. [11]

    B.Basso,L.J.DixonandA.G.Tumanov,JHEP02(2025),034doi:10.1007/JHEP02(2025)034 [arXiv:2410.22402 [hep-th]]

    work page doi:10.1007/jhep02(2025)034 2025

  12. [12]

    A.B.Goncharov,M.Spradlin,C.VerguandA.Volovich,Phys.Rev.Lett.105(2010),151605 doi:10.1103/PhysRevLett.105.151605 [arXiv:1006.5703 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.105.151605 2010

  13. [13]

    Sever, A

    A. Sever, A. G. Tumanov and M. Wilhelm, Phys. Rev. Lett.126(2021) no.3, 031602 doi:10.1103/PhysRevLett.126.031602 [arXiv:2009.11297 [hep-th]]

    work page doi:10.1103/physrevlett.126.031602 2021

  14. [14]

    Sever, A

    A. Sever, A. G. Tumanov and M. Wilhelm, JHEP10(2021), 071 doi:10.1007/JHEP10(2021)071 [arXiv:2105.13367 [hep-th]]

    work page doi:10.1007/jhep10(2021)071 2021

  15. [15]

    Sever, A

    A. Sever, A. G. Tumanov and M. Wilhelm, JHEP03(2022), 128 doi:10.1007/JHEP03(2022)128 [arXiv:2112.10569 [hep-th]]

    work page doi:10.1007/jhep03(2022)128 2022

  16. [16]

    Basso and A

    B. Basso and A. G. Tumanov, JHEP02(2024), 022 doi:10.1007/JHEP02(2024)022 [arXiv:2308.08432 [hep-th]]

    work page doi:10.1007/jhep02(2024)022 2024

  17. [17]

    L. J. Dixon, Ö. Gürdoğan, A. J. McLeod and M. Wilhelm, Phys. Rev. Lett.128(2022) no.11, 11 doi:10.1103/PhysRevLett.128.111602 [arXiv:2112.06243 [hep-th]]

    work page doi:10.1103/physrevlett.128.111602 2022

  18. [18]

    Caron-Huot, L

    S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod and G. Papathanasiou, JHEP08(2019), 016 doi:10.1007/JHEP08(2019)016 [arXiv:1903.10890 [hep-th]]

    work page doi:10.1007/jhep08(2019)016 2019

  19. [19]

    L. J. Dixon and Y.-T. Liu, JHEP09(2023), 098 doi:10.1007/JHEP09(2023)098 [arXiv:2308.08199 [hep-th]]. 10 Eight loop form factors, amplitudes and patterns in planarN=4Lance J. Dixon

    work page doi:10.1007/jhep09(2023)098 2023

  20. [20]

    The Eight-Loop Three-Point Form Factor oftr𝜙3 in PlanarN=4 Super-Yang-Mills Theory

    L. J. Dixon and Z. Li, “The Eight-Loop Three-Point Form Factor oftr𝜙3 in PlanarN=4 Super-Yang-Mills Theory”, to appear

  21. [21]

    The last of the simple remainders

    A. Brandhuber, B. Penante, G. Travaglini and C. Wen, JHEP08(2014), 100 doi:10.1007/JHEP08(2014)100 [arXiv:1406.1443 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2014)100 2014

  22. [22]

    G. Lin, G. Yang and S. Zhang, Phys. Rev. Lett.127(2021) no.17, 171602 doi:10.1103/PhysRevLett.127.171602 [arXiv:2106.05280 [hep-th]]

    work page doi:10.1103/physrevlett.127.171602 2021

  23. [23]

    G. Lin, G. Yang and S. Zhang, JHEP03(2022), 061 doi:10.1007/JHEP03(2022)061 [arXiv:2111.03021 [hep-th]]

    work page doi:10.1007/jhep03(2022)061 2022

  24. [24]

    J. M. Henn, J. Lim and W. J. Torres Bobadilla, JHEP02(2025), 085 doi:10.1007/JHEP02(2025)085 [arXiv:2410.22465 [hep-th]]

    work page doi:10.1007/jhep02(2025)085 2025

  25. [25]

    Analytic two-loop form factors in N=4 SYM

    A. Brandhuber, G. Travaglini and G. Yang, JHEP05(2012), 082 doi:10.1007/JHEP05(2012)082 [arXiv:1201.4170 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2012)082 2012

  26. [26]

    Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

    C. Duhr, JHEP08(2012), 043 doi:10.1007/JHEP08(2012)043 [arXiv:1203.0454 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2012)043 2012

  27. [27]

    X. Chen, X. Guan and B. Mistlberger, [arXiv:2504.06490 [hep-ph]]

    arXiv

  28. [28]

    L. J. Dixon, J. M. Drummond and J. M. Henn, JHEP11(2011), 023 doi:10.1007/JHEP11(2011)023 [arXiv:1108.4461 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep11(2011)023 2011

  29. [29]

    L. J. Dixon, J. M. Drummond, C. Duhr and J. Pennington, JHEP06(2014), 116 doi:10.1007/JHEP06(2014)116 [arXiv:1402.3300 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep06(2014)116 2014

  30. [30]

    J. M. Drummond, G. Papathanasiou and M. Spradlin, JHEP03(2015), 072 doi:10.1007/JHEP03(2015)072 [arXiv:1412.3763 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2015)072 2015

  31. [31]

    S. He, X. Jiang, X. Li and J. Liu, [arXiv:2511.09669 [hep-th]]

    arXiv

  32. [32]

    Caron-Huot, L

    S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod and G. Papathanasiou, JHEP09(2019), 061 doi:10.1007/JHEP09(2019)061 [arXiv:1906.07116 [hep-th]]

    work page doi:10.1007/jhep09(2019)061 2019

  33. [33]

    Cluster adjacency properties of scattering amplitudes

    J. Drummond, J. Foster and Ö. Gürdoğan, Phys. Rev. Lett.120(2018) no.16, 161601 doi:10.1103/PhysRevLett.120.161601 [arXiv:1710.10953 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.120.161601 2018

  34. [34]

    Harmonic Polylogarithms

    E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A15(2000), 725-754 doi:10.1142/S0217751X00000367 [arXiv:hep-ph/9905237 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x00000367 2000

  35. [35]

    Li,SparseRREFprime solver, at https://github.com/munuxi/SparseRREF

    Z. Li,SparseRREFprime solver, at https://github.com/munuxi/SparseRREF

  36. [36]

    N.Beisert,B.EdenandM.Staudacher,J.Stat.Mech.0701(2007),P01021doi:10.1088/1742- 5468/2007/01/P01021 [arXiv:hep-th/0610251 [hep-th]]. 11

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1742- 2007