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REVIEW 2 major objections 3 minor 61 references

This paper proves that every tree-level Yang-Mills amplitude with at least one gluon and any number of fundamental scalar or spinor matter fields is invariant under color-factor shifts, and derives the BCJ relations from this symmetry.

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2026-08-02 07:24 UTC pith:QW5GO2FM

load-bearing objection A careful, honest perturbiner proof of a known theorem — worth refereeing, with the one real risk being the not-fully-displayed bookkeeping in the S_(2,2) cancellation. the 2 major comments →

arxiv 2607.10006 v2 pith:QW5GO2FM submitted 2026-07-10 hep-th hep-ph

Color-factor symmetry using perturbiner methods for tree-level amplitudes of Yang-Mills theory coupled to matter

Abigail R. Chriss , Mia Karlsson , Stephen G. Naculich This is my paper
classification hep-th hep-ph
keywords color-factor symmetryBCJ relationscolor-kinematic dualityperturbiner expansiontree-level amplitudesYang-Mills with fundamental matterscattering amplitudes
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 establishes color-factor symmetry for all tree-level amplitudes of Yang-Mills theory coupled to fundamental matter: shifting the color factors of any amplitude that contains at least one gluon by a momentum-dependent amount leaves the amplitude unchanged. The proof is recursive, uses the perturbiner expansion as a generating function for the tree amplitudes, and covers an arbitrary number of scalar or spinor matter fields together with any number of gluons. Since the shift acts on the color factors only, invariance of the amplitude directly implies the fundamental BCJ relations among partial amplitudes, with no need to assume color-kinematic duality. The result matters because it upgrades a previously known symmetry from a case-by-case demonstration to a single all-multiplicity proof, and because the symmetry is gauge-invariant whereas color-kinematic duality is gauge-dependent.

Core claim

On the paper's own terms, the central claim is equation (5.33): the n-point tree-level amplitude A_n containing at least one gluon and any number of fundamental scalars or spinors is annihilated by the color-factor shift associated with any external gluon, δ_n A_n = 0. The derivation writes the amplitude as the residue of a color-dressed off-shell current, applies the elementary shift rules δ_a f^{bac} = α_a δ^{bc}(k_c^2 − k_b^2) and δ_a (T^a)_{mn} = α_a δ_{mn}(k_n^2 − k_m^2), and shows that the shifted current S^μ collapses to k_P^μ times a kinematical prefactor, which vanishes when contracted with the on-shell gluon polarization ε_{μ n}. Section 3 shows that this invariance, combined with

What carries the argument

The perturbiner expansion—a recursive, color-dressed solution of the classical Yang-Mills plus matter equations of motion in Lorenz gauge—acts as the generating function for all tree-level amplitudes. Auxiliary fields (G^{νμ}, Θ^μ, Υ^μ) are introduced so that the recursion relations remain at most quadratic in the fields. The amplitude is extracted as the residue of the P = 12...(n−1) current at the k_P^2 pole; applying the color-factor shift to this residue produces a current S^μ that the proof rearranges into a pure-Yang-Mills part and matter parts. The operative identity is that S^μ equals k_P^μ times a residual kinematical factor, so contraction with ε_{μ n} kills it; the matter sub-piec

Load-bearing premise

The proof assumes the perturbiner expansion enumerates every tree-level Feynman diagram in Lorenz gauge, and it imports the pure-Yang-Mills piece S^μ_YM from an earlier paper without re-derivation; if either of those fails, the conclusion δ_n A_n = 0 is not established.

What would settle it

Take a five-point amplitude with two fundamental scalars and three gluons at generic momenta, construct its color-dressed expression, apply the color-factor shift (3.2) to the gluon labels, and evaluate numerically; any nonzero shift would disprove the theorem. A complementary check is to verify that the recursive currents (4.17)-(4.26) reproduce the complete Feynman-diagram amplitude for a chosen six-point configuration.

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

If this is right

  • The fundamental BCJ relations (2.13) hold for every tree-level amplitude with at least one gluon and arbitrary numbers of fundamental scalar or spinor fields, without assuming color-kinematic duality.
  • The color-factor symmetry, and therefore the resulting BCJ relations, are gauge-invariant statements, independent of any generalized-gauge choice for kinematic numerators.
  • For the four-point matter-antimatter annihilation amplitude, the invariance forces the kinematic Jacobi identity n(1) + n(2) + n(3) = 0 directly, connecting the symmetry to color-kinematic duality at the lowest multiplicity.
  • The proof is uniform in n: because it is built from recursion relations, it establishes the symmetry for all multiplicities at once rather than instance by instance.
  • The result covers both scalar and spinor fundamental matter with arbitrary masses, under the single requirement that the amplitude contains at least one external gluon.

Where Pith is reading between the lines

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

  • The recursive structure used here may extend naturally to matter in other representations of the gauge group, or to theories with additional matter self-interactions, as long as the auxiliary-field construction keeps the equations of motion quadratic.
  • Because the proof isolates matter contributions into separately vanishing pieces, it offers a template for attacking the open question of color-factor symmetry at loop level, where color-kinematic duality is still conjectural.
  • The factorized final form—the shifted current proportional to the total word momentum—suggests a conservation-law interpretation of the symmetry that could lead to a Noether-style derivation from the classical action.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper extends the color-dressed perturbiner approach to Yang–Mills theory coupled to fundamental scalars and spinors. After deriving the recursion relations (4.17)–(4.26) for the Berends–Giele currents and the associated auxiliary fields, the authors express the n-point tree-level amplitude with at least one gluon as the residue contraction (4.37). Applying the color-factor shift (3.2) to this expression, they decompose the variation into a pure Yang–Mills contribution and a matter contribution. Using the recurrences and the auxiliary-field identities, they show that the matter contribution is proportional to k_P^μ and hence vanishes when contracted with the on-shell gluon polarization, giving δ_n A_n = 0 (Eq. (5.33)). From this invariance they derive the fundamental BCJ relations (2.13). The appendix verifies the four-point matter–antimatter annihilation amplitudes for both scalar and spinor matter, reproducing the known numerator identities.

Significance. If the proof is correct, it provides a more direct and constructive derivation of color-factor symmetry for a broad class of matter-coupled gauge-theory amplitudes, replacing the assumption of color-kinematic duality with a gauge-invariant symmetry and yielding the BCJ relations. The recursive perturbiner method is transparently organized, and the explicit four-point examples help validate the formalism. The result corroborates earlier proofs based on the radiation vertex expansion and extends the all-gluon perturbiner proof of ref. [19]. The paper is well written and the logic is largely clear. Its main weakness is that some of the most involved algebraic steps in the central cancellation are summarized rather than displayed.

major comments (2)
  1. [§5, Eqs. (5.24)–(5.27)] The crucial cancellation S^μ_(2,2)=0 is obtained by collecting the ten contributions S^μ_1...S^μ_10 into the expression (5.24), then simplifying to (5.25) and (5.26). The transition from the definitions (5.8)–(5.18) to (5.24) is not shown term by term, and the simplification from (5.24) to (5.25) is highly condensed. Since any missed or double-counted term would invalidate the central identity (5.33), the proof is not fully verifiable as written. Please provide a complete derivation of (5.24)–(5.27), either in the text or in a supplementary appendix / computer-algebra file, so the bookkeeping can be checked.
  2. [§5, Eqs. (5.28)–(5.31)] The O(g^3) cancellation S^μ_(3)=0 suffers from the same issue. The collection of terms into (5.28) and the simplification to (5.29) omit intermediate steps, and the antisymmetry argument in (5.30)–(5.31) relies on the correct retention of all terms. Please expand this part as well, or state that a machine-checkable notebook is available.
minor comments (3)
  1. [§5, Eq. (5.3)] In the spinor term, the first summand reads "−\tildeΨ^n_Q(... )γ^μ Ψ^n_R" while the original contraction in (5.1) is between \tildeΨ^m_Q and Ψ^n_R. The indices appear inconsistent; please correct.
  2. [§5] The notation for Ξ is used inconsistently, e.g. \tildeΞ^{m*}_{νB} versus \tildeΞ^{νm*}_B. Please standardize the placement of indices to improve readability.
  3. [§5, Eq. (5.6)] The pure–Yang–Mills result S^μ_YM is imported from ref. [19]. It would be helpful to add a few sentences recalling how that result is obtained in the present convention (including the sign change explained in footnote 7), so the reader does not need to consult the earlier paper for a critical input.

Circularity Check

0 steps flagged

No significant circularity: the matter-sector color-factor symmetry proof is self-contained, with the only external import being a prior independent pure-YM result.

full rationale

The paper's central claim is δ_n A_n = 0 for gluon-plus-matter amplitudes, proven in §5. This is not a renamed input or a parameter fitted to the target result: the derivation starts from the Yang-Mills-matter Lagrangian (4.1), derives the perturbiner recurrences (4.17)–(4.26), expresses amplitudes via (4.37), and reduces the color-factor-shifted amplitude to ε_n·S^μ, showing S^μ is proportional to k_P^μ so that ε_n·S^μ = 0 by ε_n·k_n = 0. The matter-sector cancellations S^μ_(2,2) = 0 in (5.27) and S^μ_(3) = 0 in (5.31) are obtained by explicit algebra using the commutation relations (2.3) and antisymmetry, not by assuming the desired invariance. The only externally imported ingredient is the pure-YM contribution S^μ_YM in (5.6), quoted from ref. [19], a same-author prior paper. This is a self-citation, but it is a parameter-free proof whose stated assumptions (pure Yang-Mills) do not contain the target matter-amplitude claim; it is therefore independent support rather than a circular restatement of the current result. The appendix also cross-checks the four-point scalar and spinor amplitudes against the Feynman-diagram expressions in (2.5)–(2.6), providing an external benchmark. The skeptic's concern about the bookkeeping from S_1...S_10 to the collected terms (5.24) and (5.29) is a possible algebra/correctness risk, not a circularity: a missed or double-counted term would be an error, not an instance of assuming the conclusion. No circular step is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

No free parameters or fitted constants. The proof relies on standard perturbiner/BCJ background and one prior result for pure YM; the only new mathematical objects are bookkeeping auxiliary fields. No new physical entities are postulated.

axioms (6)
  • domain assumption The Yang-Mills + scalar/spinor matter Lagrangian (4.1) and its equations of motion (4.3) define the amplitudes whose color-factor symmetry is being proven.
    The proof starts from the classical EOM derived from the Lagrangian and treats their perturbative solution as the generating function of tree amplitudes; this standard QFT identification is not re-derived.
  • domain assumption The color-dressed perturbiner ansatz (4.16) with ordered words P=p1...pm, p1<...<pm, together with the residue formula (4.36), generates all tree-level n-point amplitudes.
    This is the Berends-Giele/perturbiner theorem from refs [20-28]; it is load-bearing because eq (4.37) is the starting point of the proof.
  • domain assumption The pure Yang-Mills contribution S_μYM is given by eq (5.6) from the same author's prior paper [19].
    The matter-sector proof imports this result without re-derivation; if the pure-YM identity were wrong or convention-mismatched, the conclusion δ_n A_n=0 would not follow.
  • domain assumption The color-factor shift rules (3.2) define a gauge-invariant symmetry, so proving invariance in Lorenz gauge suffices for the general amplitude.
    The paper imposes ∂_μ A^μ=0 (4.5) and cites [15,16] for the gauge-invariance of CFS rather than proving it here.
  • standard math The color-ordered products of generators in the proper/Johansson-Ochirov decomposition form a linearly independent basis, so invariance under shifts implies term-by-term BCJ relations.
    Eqs. (3.3)-(3.6) require the products (T...T)_{m1m2} to be independent, a property established in the cited MJO literature [13,56,57].
  • standard math Standard Dirac algebra, the fundamental-representation commutation relations (2.3), momentum conservation, and on-shell conditions are used throughout sec. 5.
    These are ordinary algebraic identities, invoked e.g. in eqs. (5.15), (5.26), and (5.30).
invented entities (1)
  • Auxiliary fields G^{νμa}, Θ^{μn}, Υ^{μn} no independent evidence
    purpose: Rewrite the equations of motion so they are at most quadratic in fields (eqs. 4.7 and 4.8), enabling the perturbiner recursion.
    These are defined linear combinations of the original fields (eq. 4.7), not new physical degrees of freedom; they carry no independent evidence by design.

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read the original abstract

Bern-Carrasco-Johansson relations are helicity-independent constraints among the partial amplitudes of tree-level processes containing gluons and possibly also matter fields. They are a consequence of color-kinematic duality, but alternatively can be derived using the color-factor symmetry of tree-level amplitudes. We use recursive perturbiner methods to present a general, streamlined proof of the color-factor symmetry of all tree-level amplitudes containing an arbitrary number of fundamental spin-zero or spin-one-half matter fields and at least one gluon.

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Works this paper leans on

61 extracted references · 55 linked inside Pith

  1. [1]

    New Relations for Gauge-Theory Amplitudes

    Z. Bern, J. J. M. Carrasco and H. Johansson,“New Relations for Gauge-Theory Amplitudes”, Phys. Rev. D78, 085011 (2008),arxiv:0805.3993

    Pith/arXiv arXiv 2008

  2. [2]

    Perturbative Quantum Gravity as a Double Copy of Gauge Theory

    Z. Bern, J. J. M. Carrasco and H. Johansson,“Perturbative Quantum Gravity as a Double Copy of Gauge Theory”,Phys.Rev.Lett. 105, 061602 (2010),arxiv:1004.0476

    Pith/arXiv arXiv 2010

  3. [3]

    Gravity as the Square of Gauge Theory

    Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier,“Gravity as the Square of Gauge Theory”, Phys.Rev. D82, 065003 (2010),arxiv:1004.0693

    Pith/arXiv arXiv 2010

  4. [4]

    The Duality Between Color and Kinematics and its Applications

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban,“The Duality Between Color and Kinematics and its Applications”,arxiv:1909.01358

    Pith/arXiv arXiv 1909

  5. [5]

    Minimal Basis for Gauge Theory Amplitudes

    N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove,“Minimal Basis for Gauge Theory Amplitudes”,Phys. Rev. Lett. 103, 161602 (2009),arxiv:0907.1425

    Pith/arXiv arXiv 2009

  6. [6]

    Open & Closed vs. Pure Open String Disk Amplitudes

    S. Stieberger,“Open & Closed vs. Pure Open String Disk Amplitudes”,arxiv:0907.2211

    Pith/arXiv arXiv

  7. [7]

    Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program

    B. Feng, R. Huang and Y. Jia,“Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program”,Phys. Lett. B695, 350 (2011),arxiv:1004.3417

    Pith/arXiv arXiv 2011

  8. [8]

    A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory

    Y.-X. Chen, Y.-J. Du and B. Feng,“A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory”,JHEP 1102, 112 (2011),arxiv:1101.0009

    Pith/arXiv arXiv 2011

  9. [9]

    Zeros in Scattering Amplitudes and the Structure of Nonabelian Gauge Theories

    D.-p. Zhu,“Zeros in Scattering Amplitudes and the Structure of Nonabelian Gauge Theories”, Phys.Rev. D22, 2266 (1980)

    1980

  10. [10]

    Angular zeros of Brown, Mikaelian, Sahdev, and Samuel and the factorization of tree amplitudes in gauge theories

    C. Goebel, F. Halzen and J. Leveille,“Angular zeros of Brown, Mikaelian, Sahdev, and Samuel and the factorization of tree amplitudes in gauge theories”,Phys.Rev. D23, 2682 (1981)

    1981

  11. [11]

    Pure Gravities via Color-Kinematics Duality for Fundamental Matter

    H. Johansson and A. Ochirov,“Pure Gravities via Color-Kinematics Duality for Fundamental Matter”,JHEP 1511, 046 (2015),arxiv:1407.4772

    Pith/arXiv arXiv 2015

  12. [12]

    Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles

    S. G. Naculich,“Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles”,JHEP 1409, 029 (2014),arxiv:1407.7836

    Pith/arXiv arXiv 2014

  13. [13]

    Color-Kinematics Duality for QCD Amplitudes

    H. Johansson and A. Ochirov,“Color-Kinematics Duality for QCD Amplitudes”, JHEP 1601, 170 (2016),arxiv:1507.00332

    Pith/arXiv arXiv 2016

  14. [14]

    Proof of the fundamental BCJ relations for QCD amplitudes

    L. de la Cruz, A. Kniss and S. Weinzierl,“Proof of the fundamental BCJ relations for QCD amplitudes”,JHEP 1509, 197 (2015),arxiv:1508.01432

    Pith/arXiv arXiv 2015

  15. [15]

    BCJ relations from a new symmetry of gauge-theory amplitudes

    R. W. Brown and S. G. Naculich,“BCJ relations from a new symmetry of gauge-theory amplitudes”,JHEP 1610, 130 (2016),arxiv:1608.04387

    Pith/arXiv arXiv 2016

  16. [16]

    Color-factor symmetry and BCJ relations for QCD amplitudes

    R. W. Brown and S. G. Naculich,“Color-factor symmetry and BCJ relations for QCD amplitudes”,JHEP 1611, 060 (2016),arxiv:1608.05291

    Pith/arXiv arXiv 2016

  17. [17]

    KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry

    R. W. Brown and S. G. Naculich,“KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry”,JHEP 1803, 057 (2018),arxiv:1802.01620

    Pith/arXiv arXiv 2018

  18. [18]

    Classical Radiation Zeros in Gauge Theory Amplitudes

    R. W. Brown, K. Kowalski and S. J. Brodsky,“Classical Radiation Zeros in Gauge Theory Amplitudes”,Phys.Rev. D28, 624 (1983)

    1983

  19. [19]

    Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods

    S. G. Naculich,“Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods”,JHEP 2306, 084 (2023),arxiv:2304.01287. 27

    Pith/arXiv arXiv 2023

  20. [20]

    Recursive Calculations for Processes with n Gluons

    F. A. Berends and W. T. Giele,“Recursive Calculations for Processes with n Gluons”, Nucl. Phys. B 306, 759 (1988)

    1988

  21. [21]

    On amplitudes in selfdual sector of Yang-Mills theory

    A. A. Rosly and K. G. Selivanov,“On amplitudes in selfdual sector of Yang-Mills theory”, Phys. Lett. B 399, 135 (1997),hep-th/9611101

    Pith/arXiv arXiv 1997

  22. [22]

    On tree form-factors in (supersymmetric) Yang-Mills theory

    K. G. Selivanov,“On tree form-factors in (supersymmetric) Yang-Mills theory”, Commun. Math. Phys. 208, 671 (2000),hep-th/9809046

    Pith/arXiv arXiv 2000

  23. [23]

    Post-classicism in Tree Amplitudes

    K. G. Selivanov,“Post-classicism in Tree Amplitudes”,hep-th/9905128, in:“34th Rencontres de Moriond: Electroweak Interactions and Unified Theories”, 473–478p

    Pith/arXiv arXiv

  24. [24]

    Non-linear gauge transformations inD= 10SYM theory and the BCJ duality

    S. Lee, C. R. Mafra and O. Schlotterer,“Non-linear gauge transformations inD= 10SYM theory and the BCJ duality”,JHEP 1603, 090 (2016),arxiv:1510.08843

    Pith/arXiv arXiv 2016

  25. [25]

    Berends-Giele recursions and the BCJ duality in superspace and components

    C. R. Mafra and O. Schlotterer,“Berends-Giele recursions and the BCJ duality in superspace and components”,JHEP 1603, 097 (2016),arxiv:1510.08846

    Pith/arXiv arXiv 2016

  26. [26]

    Berends-Giele recursion for double-color-ordered amplitudes

    C. R. Mafra,“Berends-Giele recursion for double-color-ordered amplitudes”, JHEP 1607, 080 (2016),arxiv:1603.09731

    Pith/arXiv arXiv 2016

  27. [27]

    Berends-Giele currents in Bern-Carrasco-Johansson gauge forF 3- andF 4-deformed Yang-Mills amplitudes

    L. M. Garozzo, L. Queimada and O. Schlotterer,“Berends-Giele currents in Bern-Carrasco-Johansson gauge forF 3- andF 4-deformed Yang-Mills amplitudes”, JHEP 1902, 078 (2019),arxiv:1809.08103

    Pith/arXiv arXiv 1902

  28. [28]

    Perturbiner Methods for Effective Field Theories and the Double Copy

    S. Mizera and B. Skrzypek,“Perturbiner Methods for Effective Field Theories and the Double Copy”,JHEP 1810, 018 (2018),arxiv:1809.02096

    Pith/arXiv arXiv 2018

  29. [29]

    Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories

    T. Macrelli, C. S¨ amann and M. Wolf,“Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories”,Phys. Rev. D 100, 045017 (2019),arxiv:1903.05713

    Pith/arXiv arXiv 2019

  30. [30]

    L ∞-algebras and the perturbiner expansion

    C. Lopez-Arcos and A. Q. V´ elez,“L ∞-algebras and the perturbiner expansion”, JHEP 1911, 010 (2019),arxiv:1907.12154

    Pith/arXiv arXiv 1911

  31. [31]

    TheL ∞ structure of gauge theories with matter

    H. Gomez, R. L. Jusinskas, C. Lopez-Arcos and A. Q. Velez,“TheL ∞ structure of gauge theories with matter”,JHEP 2102, 093 (2021),arxiv:2011.09528

    Pith/arXiv arXiv 2021

  32. [32]

    Color-kinematics duality from the Bern-Kosower formalism

    N. Ahmadiniaz, F. M. Balli, C. Lopez-Arcos, A. Q. Velez and C. Schubert,“Color-kinematics duality from the Bern-Kosower formalism”,Phys. Rev. D 104, L041702 (2021), arxiv:2105.06745

    Pith/arXiv arXiv 2021

  33. [33]

    Manifest colour-kinematics duality and double-copy in the string-based formalism

    N. Ahmadiniaz, F. M. Balli, O. Corradini, C. Lopez-Arcos, A. Q. Velez and C. Schubert, “Manifest colour-kinematics duality and double-copy in the string-based formalism”, Nucl. Phys. B 975, 115690 (2022),arxiv:2110.04853

    Pith/arXiv arXiv 2022

  34. [34]

    Multiparticle Solutions to Einstein ’s Equations

    H. Gomez and R. L. Jusinskas,“Multiparticle Solutions to Einstein ’s Equations”, Phys. Rev. Lett. 127, 181603 (2021),arxiv:2106.12584

    Pith/arXiv arXiv 2021

  35. [35]

    New recursion relations for tree-level correlators in anti–de Sitter spacetime

    C. Armstrong, H. Gomez, R. Lipinski Jusinskas, A. Lipstein and J. Mei,“New recursion relations for tree-level correlators in anti–de Sitter spacetime”,Phys. Rev. D 106, L121701 (2022), arxiv:2209.02709

    Pith/arXiv arXiv 2022

  36. [36]

    One-Loop Off-Shell Amplitudes from Classical Equations of Motion

    H. Gomez, R. Lipinski Jusinskas, C. Lopez-Arcos and A. Quintero Velez,“One-Loop Off-Shell Amplitudes from Classical Equations of Motion”,Phys. Rev. Lett. 130, 081601 (2023), arxiv:2208.02831. 28

    Pith/arXiv arXiv 2023

  37. [37]

    Loop Amplitudes and Quantum Homotopy Algebras

    B. Jurˇ co, T. Macrelli, C. S¨ amann and M. Wolf,“Loop Amplitudes and Quantum Homotopy Algebras”,JHEP 2007, 003 (2020),arxiv:1912.06695

    Pith/arXiv arXiv 2007

  38. [38]

    Double Copy from Homotopy Algebras

    L. Borsten, H. Kim, B. Jurˇ co, T. Macrelli, C. Saemann and M. Wolf,“Double Copy from Homotopy Algebras”,Fortsch. Phys. 69, 2100075 (2021),arxiv:2102.11390

    Pith/arXiv arXiv 2021

  39. [39]

    Homotopy double copy and the Kawai–Lewellen–Tye relations for the non-abelian and tensor Navier–Stokes equations

    V. G. Escudero, C. Lopez-Arcos and A. Quintero Velez,“Homotopy double copy and the Kawai–Lewellen–Tye relations for the non-abelian and tensor Navier–Stokes equations”, J. Math. Phys. 64, 2881598 (2023),arxiv:2201.06047

    Pith/arXiv arXiv 2023

  40. [40]

    Colour-kinematics duality, double copy, and homotopy algebras

    L. Borsten, H. Kim, B. Jurco, T. Macrelli, C. Saemann and M. Wolf,“Colour-kinematics duality, double copy, and homotopy algebras”,PoS ICHEP2022, 426 (2022),arxiv:2211.16405

    Pith/arXiv arXiv 2022

  41. [41]

    The off-shell recursion for gravity and the classical double copy for currents

    K. Cho, K. Kim and K. Lee,“The off-shell recursion for gravity and the classical double copy for currents”,JHEP 2201, 186 (2022),arxiv:2109.06392

    Pith/arXiv arXiv 2022

  42. [42]

    Quantum off-shell recursion relation

    K. Lee,“Quantum off-shell recursion relation”,JHEP 2205, 051 (2022),arxiv:2202.08133

    Pith/arXiv arXiv 2022

  43. [43]

    Perturbations of general relativity to all orders and the general n th order terms

    K. Cho, K. Kim and K. Lee,“Perturbations of general relativity to all orders and the general n th order terms”,JHEP 2303, 112 (2023),arxiv:2209.11424

    Pith/arXiv arXiv 2023

  44. [44]

    Binary black holes and quantum off-shell recursion

    K. Cho, K. Kim and K. Lee,“Binary black holes and quantum off-shell recursion”, JHEP 2405, 050 (2024),arxiv:2311.01284

    Pith/arXiv arXiv 2024

  45. [45]

    Schwarzschild Black Hole from Perturbation Theory to All Orders

    P. H. Damgaard and K. Lee,“Schwarzschild Black Hole from Perturbation Theory to All Orders”,Phys. Rev. Lett. 132, 251603 (2024),arxiv:2403.13216

    Pith/arXiv arXiv 2024

  46. [46]

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

    Y.-X. Tao,“Systematic approach toℓ-loop planar integrands from the classical equation of motion”,Phys. Rev. D 112, 106002 (2025),arxiv:2504.15701

    Pith/arXiv arXiv 2025

  47. [47]

    The bi-adjoint scalarℓ-loop planar integrand recursion and graded inverse variables

    Y.-X. Tao,“The bi-adjoint scalarℓ-loop planar integrand recursion and graded inverse variables”, JHEP 2603, 165 (2026),arxiv:2505.07077

    Pith/arXiv arXiv 2026

  48. [48]

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

    Y.-X. Tao,“Off-shell recursion for all-loop planar integrands in Yang-Mills theory”, arxiv:2604.20377

    Pith/arXiv arXiv

  49. [49]

    Perturbiner methods in scattering amplitude

    R. Lipinski Jusinskas,“Perturbiner methods in scattering amplitude”,arxiv:2607.06661

    Pith/arXiv arXiv

  50. [50]

    Multi-Gluon Scattering: A String Based Calculation

    D. Kosower, B.-H. Lee and V. P. Nair,“Multi-Gluon Scattering: A String Based Calculation”, Phys. Lett. B201, 85 (1988)

    1988

  51. [51]

    The Color Structure of Gluon Emission

    M. L. Mangano,“The Color Structure of Gluon Emission”,Nucl. Phys. B309, 461 (1988)

    1988

  52. [52]

    Perturbative Gravity and Gauge Theory Relations: A Review

    T. Sondergaard,“Perturbative Gravity and Gauge Theory Relations: A Review”, Adv. High Energy Phys. 2012, 726030 (2012),arxiv:1106.0033

    Pith/arXiv arXiv 2012

  53. [53]

    One loop selfdual and N=4 superYang-Mills

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,“One loop selfdual and N=4 superYang-Mills”,Phys.Lett. B394, 105 (1997),hep-th/9611127

    Pith/arXiv arXiv 1997

  54. [54]

    Recursion relations for gauge theory amplitudes with massive particles

    S. Badger, E. N. Glover, V. Khoze and P. Svrcek,“Recursion relations for gauge theory amplitudes with massive particles”,JHEP 0507, 025 (2005),hep-th/0504159

    Pith/arXiv arXiv 2005

  55. [55]

    All-multiplicity amplitudes with massive scalars

    D. Forde and D. A. Kosower,“All-multiplicity amplitudes with massive scalars”, Phys.Rev. D73, 065007 (2006),hep-th/0507292

    Pith/arXiv arXiv 2006

  56. [56]

    Dyck words and multiquark primitive amplitudes

    T. Melia,“Dyck words and multiquark primitive amplitudes”,Phys. Rev. D88, 014020 (2013), arxiv:1304.7809. 29

    Pith/arXiv arXiv 2013

  57. [57]

    Getting more flavor out of one-flavor QCD

    T. Melia,“Getting more flavor out of one-flavor QCD”,Phys. Rev. D89, 074012 (2014), arxiv:1312.0599

    Pith/arXiv arXiv 2014

  58. [58]

    The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory

    Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban,“The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory”,Phys. Rev. D82, 125040 (2010), arxiv:1008.3327

    Pith/arXiv arXiv 2010

  59. [59]

    Covariant color-kinematics duality

    C. Cheung and J. Mangan,“Covariant color-kinematics duality”,JHEP 2111, 069 (2021), arxiv:2108.02276

    Pith/arXiv arXiv 2021

  60. [60]

    Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes

    N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove,“Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes”,JHEP 1006, 003 (2010),arxiv:1003.2403

    Pith/arXiv arXiv 2010

  61. [61]

    Dual Identities inside the Gluon and the Graviton Scattering Amplitudes

    S. H. Henry Tye and Y. Zhang,“Dual Identities inside the Gluon and the Graviton Scattering Amplitudes”,JHEP 1006, 071 (2010),arxiv:1003.1732, [Erratum: JHEP04,114(2011)]. 30

    Pith/arXiv arXiv 2010