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arxiv: 2605.16034 · v1 · pith:5YIVSENTnew · submitted 2026-05-15 · ✦ hep-th · hep-ph

Walking Sudakov: From Cusp to Octagon

Luis F. Alday , Elisabetta Armanini , Andrei V. Belitsky , Kelian H\"aring , Alexander Zhiboedov This is my paper

Pith reviewed 2026-05-20 17:28 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords Sudakov form factorwalking anomalous dimensioncusp anomalous dimensionoctagon anomalous dimensionCoulomb branchplanar N=4 SYMinfrared divergencesdouble logarithms
0
0 comments X

The pith

A walking anomalous dimension interpolates between the cusp and octagon regimes for Sudakov form factors and four-point amplitudes in planar N=4 SYM.

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

The paper examines the Sudakov form factor and four-point scattering amplitude on the Coulomb branch of planar N=4 super Yang-Mills theory. It identifies a scaling limit in which both quantities develop double-logarithmic infrared divergences controlled by a new walking anomalous dimension. As Coulomb-branch mass parameters vary, this dimension smoothly connects the standard cusp anomalous dimension of the on-shell regime to the octagon anomalous dimension of the off-shell regime. From the explicit two-loop computation and the presumed all-order structure, the authors write down candidate closed-form expressions for the walking dimension at any loop order, expressed in terms of new unknown functions of the coupling constant.

Core claim

In a controlled scaling limit, the Sudakov form factor and the four-point scattering amplitude on the Coulomb branch exhibit double-logarithmic behavior governed by a walking anomalous dimension. This quantity interpolates between the cusp anomalous dimension in the on-shell limit and the octagon anomalous dimension in the off-shell limit as the mass scales are varied. Based on the explicit two-loop result together with the expected all-order structure, closed all-loop expressions are proposed for the walking anomalous dimension of both observables; these expressions involve new, presently unknown functions of the 't Hooft coupling.

What carries the argument

The walking anomalous dimension, which controls the double-logarithmic infrared behavior and interpolates between the cusp and octagon anomalous dimensions as Coulomb-branch parameters are varied.

If this is right

  • The proposed expressions unify the description of infrared logarithms across on-shell and off-shell kinematic regimes.
  • The new unknown functions of the coupling must be fixed by higher-loop calculations or other consistency conditions.
  • The same interpolation pattern is expected to hold for other infrared-sensitive observables in the same theory.

Where Pith is reading between the lines

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

  • Analogous walking anomalous dimensions may exist for higher-point amplitudes or in other planar gauge theories with similar Coulomb-branch deformations.
  • Determining the unknown coupling-dependent functions could expose hidden integrability structures that govern the transition between different infrared regimes.
  • The scaling limit identified here offers a practical setting for testing all-order resummation techniques that bridge perturbative and non-perturbative regimes.

Load-bearing premise

That the explicit two-loop result together with an assumed all-order structure is enough to write down a closed all-loop form for the walking anomalous dimension.

What would settle it

An explicit three-loop computation of the walking anomalous dimension that fails to match the proposed all-loop expression would disprove the conjecture.

read the original abstract

We study the Sudakov form factor and the four-point scattering amplitude on the Coulomb branch of planar $\mathcal{N}=4$ SYM as functions of the Coulomb-branch parameters and kinematic invariants. This setup provides a controlled probe of the interpolation between on- and off-shell regimes of infrared-sensitive quantities in gauge theories. We identify a novel scaling limit in which both observables exhibit double-logarithmic behavior governed by a walking anomalous dimension. As the mass scales are varied, this walking anomalous dimension interpolates between the cusp anomalous dimension of the on-shell regime and the octagon anomalous dimension of the off-shell regime. Based on the explicit two-loop result and the expected all-order structure, we propose an all-loop form for the walking anomalous dimension both for the form factor and for the four-point scattering amplitude. These all-loop expressions depend on new, presently unknown functions of the 't Hooft coupling.

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 studies the Sudakov form factor and four-point scattering amplitude in planar N=4 SYM on the Coulomb branch, identifying a novel scaling limit in which both quantities exhibit double-logarithmic behavior controlled by a walking anomalous dimension. This dimension interpolates between the cusp anomalous dimension (on-shell regime) and the octagon anomalous dimension (off-shell regime) as Coulomb-branch mass scales are varied. Explicit two-loop results are presented, and an all-loop form is proposed for the walking anomalous dimension in both observables; these expressions depend on new, presently unknown functions of the 't Hooft coupling.

Significance. If the proposed all-loop interpolation holds, the result would provide a controlled probe of the transition between on-shell and off-shell infrared regimes for Sudakov-type quantities, potentially illuminating the structure of anomalous dimensions in integrable gauge theories. The explicit two-loop computation is a concrete strength that supplies a benchmark for future checks. The significance is reduced by the absence of an independent derivation or higher-loop verification for the assumed persistence of the interpolating structure.

major comments (2)
  1. [all-loop proposal section] The central all-loop proposal (abstract and the section introducing the walking anomalous dimension) rests on the explicit two-loop result combined with an 'expected all-order structure,' yet no derivation from Ward identities, integrability, or other first-principles constraints is supplied to show that the same interpolating functional form continues to govern the double-logarithmic scaling at higher orders. The introduction of new unknown functions of the 't Hooft coupling therefore leaves the all-loop claim dependent on an unverified extrapolation assumption.
  2. [discussion of interpolation] The claim that the walking anomalous dimension interpolates between cusp and octagon regimes at all loops is load-bearing for the manuscript's main result, but the two-loop match alone does not constrain possible loop-dependent corrections that could modify or break the assumed functional structure beyond two loops; an explicit argument ruling out such terms is missing.
minor comments (1)
  1. [notation and definitions] The definition of the walking anomalous dimension and its relation to the standard cusp and octagon quantities would benefit from a single dedicated equation that makes the interpolation parameter explicit.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the two-loop computation. Our all-loop proposal is presented as a conjecture motivated by the explicit two-loop results and the expected all-order structure in planar N=4 SYM; we address the concerns about the extrapolation and its assumptions below. Revisions have been made to clarify the conjectural status and to discuss potential limitations.

read point-by-point responses

  1. Referee: The central all-loop proposal (abstract and the section introducing the walking anomalous dimension) rests on the explicit two-loop result combined with an 'expected all-order structure,' yet no derivation from Ward identities, integrability, or other first-principles constraints is supplied to show that the same interpolating functional form continues to govern the double-logarithmic scaling at higher orders. The introduction of new unknown functions of the 't Hooft coupling therefore leaves the all-loop claim dependent on an unverified extrapolation assumption.

    Authors: We agree that the all-loop form is an extrapolation from the two-loop data combined with the anticipated all-order structure, without a derivation from Ward identities or integrability at this stage. The new coupling-dependent functions are introduced because their explicit expressions are not determined by the present calculation. In the revised manuscript we have updated the abstract and the section on the walking anomalous dimension to state explicitly that the interpolating expression is conjectural, and we have added a paragraph discussing the structural motivation for the assumed persistence of the form. revision: yes

  2. Referee: The claim that the walking anomalous dimension interpolates between cusp and octagon regimes at all loops is load-bearing for the manuscript's main result, but the two-loop match alone does not constrain possible loop-dependent corrections that could modify or break the assumed functional structure beyond two loops; an explicit argument ruling out such terms is missing.

    Authors: The two-loop agreement alone does not rigorously exclude possible loop-dependent modifications at higher orders. Our expectation that the interpolating structure persists rests on the kinematics of the scaling limit and consistency with known results for the cusp and octagon anomalous dimensions, but we do not possess an explicit argument that rules out all such corrections. We have added a caveat in the discussion section acknowledging this assumption and identifying its verification as a topic for future work. revision: partial

standing simulated objections not resolved
  • A first-principles derivation of the all-loop interpolating form from Ward identities, integrability, or other constraints.

Circularity Check

0 steps flagged

No circularity: all-loop proposal is explicitly an extrapolation from two-loop data plus assumed structure, with unknown functions left open.

full rationale

The paper does not claim a first-principles derivation or closed-form result that reduces to its inputs by construction. The abstract states the all-loop expressions are proposed 'based on the explicit two-loop result and the expected all-order structure' and 'depend on new, presently unknown functions of the 't Hooft coupling.' This is an informed ansatz/extrapolation rather than a tautological redefinition or a fitted parameter renamed as a prediction. No self-citation load-bearing step, uniqueness theorem, or renaming of a known result is invoked to force the form. The central claim remains independent of its two-loop input in the sense required by the circularity criteria; concerns about persistence beyond two loops are matters of correctness and verification, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The proposal relies on extrapolating an all-order structure from two-loop data and on introducing new undetermined functions of the coupling; these elements are not derived from first principles within the abstract and represent the main additions beyond prior cusp and octagon results.

free parameters (1)
  • new unknown functions of the 't Hooft coupling
    The all-loop expressions for the walking anomalous dimension are stated to depend on these functions, which are not computed or fixed in the abstract.
axioms (1)
  • domain assumption The two-loop result plus expected all-order structure suffices to propose the all-loop walking anomalous dimension form
    Invoked to justify the extrapolation from two loops to all orders while preserving the interpolation between cusp and octagon regimes.

pith-pipeline@v0.9.0 · 5695 in / 1571 out tokens · 56443 ms · 2026-05-20T17:28:37.785094+00:00 · methodology

discussion (0)

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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 echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    walking anomalous dimension ... interpolates between the cusp anomalous dimension ... and the octagon anomalous dimension ... Γ_walk(η,g) = Γ_cusp(g)/4 + γ(g)η + (Γ_oct(g)/2 − Γ_cusp(g)/4 − γ(g)) η² for 0≤η≤1

  • IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    octagon anomalous dimension of the off-shell regime ... 8-tick period implicit in octagon

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.

Reference graph

Works this paper leans on

104 extracted references · 104 canonical work pages · 46 internal anchors

  1. [1]

    A. H. Mueller,Perturbative QCD at High-Energies,Phys. Rept.73(1981) 237

    work page 1981

  2. [2]

    Sen,Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories,Phys

    A. Sen,Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories,Phys. Rev. D28(1983) 860

    work page 1983

  3. [3]

    J. C. Collins, D. E. Soper and G. F. Sterman,Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production,Nucl. Phys. B250(1985) 199–224

    work page 1985

  4. [4]

    J. C. Collins, D. E. Soper and G. F. Sterman,Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys.5(1989) 1–91, [hep-ph/0409313]

    work page internal anchor Pith review Pith/arXiv arXiv 1989

  5. [5]

    J. C. Collins,Sudakov form-factors,Adv. Ser. Direct. High Energy Phys.5(1989) 573–614, [hep-ph/0312336]

    work page arXiv 1989

  6. [6]

    G. F. Sterman,Partons, factorization and resummation, TASI 95, inTheoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, pp. 327–408, 6, 1995.hep-ph/9606312

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  7. [7]

    Agarwal, L

    N. Agarwal, L. Magnea, C. Signorile-Signorile and A. Tripathi,The infrared structure of perturbative gauge theories,Phys. Rept.994(2023) 1–120, [2112.07099]

    work page arXiv 2023

  8. [8]

    V. V. Sudakov,Vertex parts at very high-energies in quantum electrodynamics,Sov. Phys. JETP3(1956) 65–71

    work page 1956

  9. [9]

    Jackiw,Dynamics at high momentum and the vertex function of spinor electrodynamics, Annals Phys.48(1968) 292–321

    R. Jackiw,Dynamics at high momentum and the vertex function of spinor electrodynamics, Annals Phys.48(1968) 292–321

    work page 1968

  10. [10]

    P. M. Fishbane and J. D. Sullivan,Asymptotic behavior of the vertex function in quantum electrodynamics,Phys. Rev. D4(1971) 458–475

    work page 1971

  11. [11]

    J. C. Collins and D. E. Soper,Back-To-Back Jets in QCD,Nucl. Phys. B193(1981) 381

    work page 1981

  12. [12]

    Hard-Soft-Collinear Factorization to All Orders

    I. Feige and M. D. Schwartz,Hard-Soft-Collinear Factorization to All Orders,Phys. Rev. D 90(2014) 105020, [1403.6472]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  13. [13]

    Ma,A Forest Formula to Subtract Infrared Singularities in Amplitudes for Wide-angle Scattering,JHEP05(2020) 012, [1910.11304]

    Y. Ma,A Forest Formula to Subtract Infrared Singularities in Amplitudes for Wide-angle Scattering,JHEP05(2020) 012, [1910.11304]

    work page arXiv 2020

  14. [14]

    Ma,All-order prescription for facet regions in massless wide-angle scattering, 2601.22144

    Y. Ma,All-order prescription for facet regions in massless wide-angle scattering, 2601.22144

    work page arXiv

  15. [15]

    Botts and G

    J. Botts and G. F. Sterman,Hard Elastic Scattering in QCD: Leading Behavior,Nucl. Phys. B325(1989) 62–100

    work page 1989

  16. [16]

    M. L. Mangano and S. J. Parke,Multiparton amplitudes in gauge theories,Phys. Rept.200 (1991) 301–367, [hep-th/0509223]

    work page internal anchor Pith review Pith/arXiv arXiv 1991

  17. [17]

    Z. Bern, L. J. Dixon and D. A. Kosower,Progress in one loop QCD computations,Ann. Rev. Nucl. Part. Sci.46(1996) 109–148, [hep-ph/9602280]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  18. [18]

    Resummation for QCD Hard Scattering

    N. Kidonakis and G. F. Sterman,Resummation for QCD hard scattering,Nucl. Phys. B 505(1997) 321–348, [hep-ph/9705234]. – 38 –

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  19. [19]

    Soft radiation in heavy-particle pair production: all-order colour structure and two-loop anomalous dimension

    M. Beneke, P. Falgari and C. Schwinn,Soft radiation in heavy-particle pair production: All-order colour structure and two-loop anomalous dimension,Nucl. Phys. B828(2010) 69–101, [0907.1443]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [20]

    Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories

    A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories,JHEP11(2009) 062, [0908.3676]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  21. [21]

    Scattering Amplitudes

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

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    Bassetto, M

    A. Bassetto, M. Ciafaloni and G. Marchesini,Jet Structure and Infrared Sensitive Quantities in Perturbative QCD,Phys. Rept.100(1983) 201–272

    work page 1983

  23. [23]

    G. F. Sterman,Summation of Large Corrections to Short Distance Hadronic Cross-Sections,Nucl. Phys. B281(1987) 310–364

    work page 1987

  24. [24]

    Sudakov Factorization and Resummation

    H. Contopanagos, E. Laenen and G. F. Sterman,Sudakov factorization and resummation, Nucl. Phys. B484(1997) 303–330, [hep-ph/9604313]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  25. [25]

    The Singular Behaviour of QCD Amplitudes at Two-loop Order

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  26. [26]

    G. F. Sterman and M. E. Tejeda-Yeomans,Multiloop amplitudes and resummation,Phys. Lett. B552(2003) 48–56, [hep-ph/0210130]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  27. [27]

    Z. Bern, L. J. Dixon and V. A. Smirnov,Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond,Phys. Rev. D72(2005) 085001, [hep-th/0505205]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  28. [28]

    S. M. Aybat, L. J. Dixon and G. F. Sterman,The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole,Phys. Rev. D74(2006) 074004, [hep-ph/0607309]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  29. [29]

    L. J. Dixon, L. Magnea and G. F. Sterman,Universal structure of subleading infrared poles in gauge theory amplitudes,JHEP08(2008) 022, [0805.3515]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  30. [30]

    On the Structure of Infrared Singularities of Gauge-Theory Amplitudes

    T. Becher and M. Neubert,On the Structure of Infrared Singularities of Gauge-Theory Amplitudes,JHEP06(2009) 081, [0903.1126]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  31. [31]

    Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes

    E. Gardi and L. Magnea,Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes,JHEP03(2009) 079, [0901.1091]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  32. [32]

    L. J. Dixon, E. Gardi and L. Magnea,On soft singularities at three loops and beyond, JHEP02(2010) 081, [0910.3653]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  33. [33]

    Three-loop corrections to the soft anomalous dimension in multi-leg scattering

    Ø. Almelid, C. Duhr and E. Gardi,Three-loop corrections to the soft anomalous dimension in multileg scattering,Phys. Rev. Lett.117(2016) 172002, [1507.00047]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    Gardi and Z

    E. Gardi and Z. Zhu,Infrared singularities of multileg amplitudes with a massive particle at three loops,2510.27567

    work page arXiv

  35. [35]

    A. H. Mueller,On the Asymptotic Behavior of the Sudakov Form-factor,Phys. Rev. D20 (1979) 2037

    work page 1979

  36. [36]

    J. C. Collins,Algorithm to Compute Corrections to the Sudakov Form-factor,Phys. Rev. D 22(1980) 1478. – 39 –

    work page 1980

  37. [37]

    G. P. Korchemsky and A. V. Radyushkin,Infrared Asymptotics of Perturbative QCD. Vertex Functions,Sov. J. Nucl. Phys.45(1987) 910

    work page 1987

  38. [38]

    G. P. Korchemsky,Sudakov Form-factor in QCD,Phys. Lett. B220(1989) 629–634

    work page 1989

  39. [39]

    Magnea and G

    L. Magnea and G. F. Sterman,Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D42(1990) 4222–4227

    work page 1990

  40. [40]

    Analytic resummation for the quark form factor in QCD

    L. Magnea,Analytic resummation for the quark form-factor in QCD,Nucl. Phys. B593 (2001) 269–288, [hep-ph/0006255]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  41. [41]

    S. V. Ivanov, G. P. Korchemsky and A. V. Radyushkin,Infrared Asymptotics of Perturbative QCD: Contour Gauges,Yad. Fiz.44(1986) 230–240

    work page 1986

  42. [42]

    G. P. Korchemsky and A. V. Radyushkin,Infrared asymptotics of perturbative QCD: renormalization properties of the Wilson loops in higher orders of perturbation theory,Sov. J. Nucl. Phys.44(1986) 877

    work page 1986

  43. [43]

    G. P. Korchemsky and A. V. Radyushkin,Loop Space Formalism and Renormalization Group for the Infrared Asymptotics of QCD,Phys. Lett. B171(1986) 459–467

    work page 1986

  44. [44]

    A. M. Polyakov,Gauge Fields as Rings of Glue,Nucl. Phys. B164(1980) 171–188

    work page 1980

  45. [45]

    V. S. Dotsenko and S. N. Vergeles,Renormalizability of Phase Factors in the Nonabelian Gauge Theory,Nucl. Phys. B169(1980) 527–546

    work page 1980

  46. [46]

    G. P. Korchemsky and A. V. Radyushkin,Renormalization of the Wilson Loops Beyond the Leading Order,Nucl. Phys. B283(1987) 342–364

    work page 1987

  47. [47]

    Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production

    N. Kidonakis,Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production,Phys. Rev. Lett.102(2009) 232003, [0903.2561]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  48. [48]

    The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions

    A. Grozin, J. M. Henn, G. P. Korchemsky and P. Marquard,The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions,JHEP01(2016) 140, [1510.07803]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  49. [49]

    Kidonakis,Four-loop massive cusp anomalous dimension in QCD: A calculation from asymptotics,Phys

    N. Kidonakis,Four-loop massive cusp anomalous dimension in QCD: A calculation from asymptotics,Phys. Rev. D107(2023) 054006, [2301.05972]

    work page arXiv 2023

  50. [50]

    Grozin,QCD cusp anomalous dimension: current status,Int

    A. Grozin,QCD cusp anomalous dimension: current status,Int. J. Mod. Phys.38(2023) , [2212.05290]

    work page arXiv 2023

  51. [51]

    Brink, J

    L. Brink, J. H. Schwarz and J. Scherk,Supersymmetric Yang-Mills Theories,Nucl. Phys. B 121(1977) 77–92

    work page 1977

  52. [52]

    Gliozzi, J

    F. Gliozzi, J. Scherk and D. I. Olive,Supersymmetry, Supergravity Theories and the Dual Spinor Model,Nucl. Phys. B122(1977) 253–290

    work page 1977

  53. [53]

    Mandelstam,Light Cone Superspace and the Ultraviolet Finiteness of theN= 4Model, Nucl

    S. Mandelstam,Light Cone Superspace and the Ultraviolet Finiteness of theN= 4Model, Nucl. Phys. B213(1983) 149–168

    work page 1983

  54. [54]

    L. F. Alday, J. M. Henn, J. Plefka and T. Schuster,Scattering into the fifth dimension of N= 4super Yang-Mills,JHEP01(2010) 077, [0908.0684]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  55. [55]

    Caron-Huot and F

    S. Caron-Huot and F. Coronado,Ten dimensional symmetry ofN= 4 SYM correlators, JHEP03(2022) 151, [2106.03892]. – 40 –

    work page arXiv 2022

  56. [56]

    L. V. Bork, N. B. Muzhichkov and E. S. Sozinov,Infrared properties of five-point massive amplitudes inN= 4 SYM on the Coulomb branch,JHEP08(2022) 173, [2201.08762]

    work page arXiv 2022

  57. [57]

    A. V. Belitsky, L. V. Bork, A. F. Pikelner and V. A. Smirnov,Exact Off Shell Sudakov Form Factor inN= 4Supersymmetric Yang-Mills Theory,Phys. Rev. Lett.130(2023) 091605, [2209.09263]

    work page arXiv 2023

  58. [58]

    A. V. Belitsky, L. V. Bork and V. A. Smirnov,Off-shell form factor inN= 4sYM at three loops,JHEP11(2023) 111, [2306.16859]

    work page arXiv 2023

  59. [59]

    Mandelstam,Analytic properties of transition amplitudes in perturbation theory,Phys

    S. Mandelstam,Analytic properties of transition amplitudes in perturbation theory,Phys. Rev.115(1959) 1741–1751

    work page 1959

  60. [60]

    Coronado,Bootstrapping the Simplest Correlator in PlanarN= 4Supersymmetric Yang-Mills Theory to All Loops,Phys

    F. Coronado,Bootstrapping the Simplest Correlator in PlanarN= 4Supersymmetric Yang-Mills Theory to All Loops,Phys. Rev. Lett.124(2020) 171601, [1811.03282]

    work page arXiv 2020

  61. [61]

    A. V. Belitsky and G. P. Korchemsky,Exact null octagon,JHEP05(2020) 070, [1907.13131]

    work page arXiv 2020

  62. [62]

    A. V. Belitsky,Null octagon from Deift-Zhou steepest descent,Nucl. Phys. B980(2022) 115844, [2012.10446]

    work page arXiv 2022

  63. [63]

    J. M. Henn, S. Moch and S. G. Naculich,Form factors and scattering amplitudes inN= 4 SYM in dimensional and massive regularizations,JHEP12(2011) 024, [1109.5057]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  64. [64]

    A. V. Belitsky, L. V. Bork, R. N. Lee, A. I. Onishchenko and V. A. Smirnov,Five W-boson amplitude is equal to near-null decagon,Phys. Rev. D113(2026) 086003, [2510.16471]

    work page arXiv 2026

  65. [65]

    R. M. Schabinger,Scattering on the moduli space of n = 4 super yang-mills,0801.1542

    work page internal anchor Pith review Pith/arXiv arXiv

  66. [66]

    Massive amplitudes on the Coulomb branch of N=4 SYM

    N. Craig, H. Elvang, M. Kiermaier and T. R. Slatyer,Massive amplitudes on the coulomb branch of n = 4 sym,JHEP12(2011) 097, [1104.2050]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  67. [67]

    Herderschee, S

    A. Herderschee, S. Koren and T. Trott,Constructing n = 4 coulomb branch superamplitudes,JHEP08(2019) 107, [1902.07205]

    work page arXiv 2019

  68. [68]

    The cusp anomalous dimension at three loops and beyond

    D. Correa, J. Henn, J. Maldacena and A. Sever,The cusp anomalous dimension at three loops and beyond,JHEP05(2012) 098, [1203.1019]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  69. [69]

    The quark anti-quark potential and the cusp anomalous dimension from a TBA equation

    D. Correa, J. Maldacena and A. Sever,The quark anti-quark potential and the cusp anomalous dimension from a TBA equation,JHEP08(2012) 134, [1203.1913]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  70. [70]

    Quark--anti-quark potential in N=4 SYM

    N. Gromov and F. Levkovich-Maslyuk,Quark-anti-quark potential inN=4 SYM,JHEP 12(2016) 122, [1601.05679]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  71. [71]

    Asymptotic expansion of Feynman integrals near threshold

    M. Beneke and V. A. Smirnov,Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522(1998) 321–344, [hep-ph/9711391]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  72. [72]

    Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory

    C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower,Planar amplitudes in maximally supersymmetric Yang-Mills theory,Phys. Rev. Lett.91(2003) 251602, [hep-th/0309040]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  73. [73]

    Fronsdal and R

    C. Fronsdal and R. Norton,Integral representations for vertex functions,Journal of Mathematical Physics5(1964) 100–108

    work page 1964

  74. [74]

    R. E. Norton,Locations of Landau Singularities,Phys. Rev.135(1964) B1381–B1388. – 41 –

    work page 1964

  75. [75]

    A. V. Belitsky and V. A. Smirnov,Tropical regions of near mass-shell pentabox,Phys. Rev. D113(2026) 045019, [2508.14298]

    work page arXiv 2026

  76. [76]

    Salvatori,The Tropical Geometry of Subtraction Schemes,2406.14606

    G. Salvatori,The Tropical Geometry of Subtraction Schemes,2406.14606

    work page arXiv

  77. [77]

    On hyperlogarithms and Feynman integrals with divergences and many scales

    E. Panzer,On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP03(2014) 071, [1401.4361]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  78. [78]

    E. R. Speer,Analytic Renormalization,J. Math. Phys.9(1968) 1404

    work page 1968

  79. [79]

    A. V. Smirnov, N. D. Shapurov and L. I. Vysotsky,FIESTA5: Numerical high-performance Feynman integral evaluation,Comput. Phys. Commun.277(2022) 108386, [2110.11660]

    work page arXiv 2022

  80. [80]

    J. C. Collins and F. V. Tkachov,Breakdown of dimensional regularization in the Sudakov problem,Phys. Lett. B294(1992) 403–411, [hep-ph/9208209]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

Showing first 80 references.