Pith. sign in

REVIEW 4 minor 53 references

Tree-level Yang-Mills amplitudes with fundamental matter stay invariant under a color-factor shift on any gluon, yielding BCJ relations by recursion.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 01:04 UTC pith:QW5GO2FM

load-bearing objection Clean recursive proof that color-factor symmetry holds for tree-level YM+fundamental matter; solid technical extension of the 2023 pure-YM result, not a new physical claim.

arxiv 2607.10006 v1 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 PACS 11.15.Bt11.55.Bq12.38.Bx
keywords color-factor symmetryBCJ relationsperturbiner methodsYang-Mills matter amplitudescolor-kinematic dualityBerends-Giele currentsfundamental matter
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.

This paper proves that every tree-level scattering amplitude built from gluons plus any number of fundamental scalars or spin-1/2 fermions is unchanged when a particular momentum-dependent shift is applied to its color factors, provided at least one gluon is present. The shift is defined locally on structure constants and generators and acts differently on each diagram according to the momenta flowing through the other legs of the vertex that carries the external gluon. Because the full amplitude is invariant while the color factors transform, the kinematic partial amplitudes must themselves satisfy a linear relation known as the fundamental BCJ relation. The proof is obtained by solving the classical Yang-Mills-matter equations recursively with a color-dressed perturbiner expansion and then verifying that the resulting recursive expression for the amplitude cancels term by term under the shift. The result supplies a transparent, all-multiplicity demonstration that color-factor symmetry holds for matter-coupled theories and therefore that the BCJ constraints follow without assuming color-kinematic duality a priori.

Core claim

Every tree-level amplitude containing an arbitrary number of fundamental spin-0 or spin-1/2 matter fields and at least one gluon is invariant under the color-factor shift associated with that gluon. The invariance is established by substituting the recursive perturbiner solution of the classical equations of motion into the shift and showing that every contribution cancels after use of the Jacobi identity and momentum conservation, immediately implying the fundamental BCJ relations among the partial amplitudes.

What carries the argument

Color-dressed perturbiner expansion: a recursive solution of the classical Yang-Mills-matter equations that generates color-dressed Berends-Giele currents; the n-point amplitude is recovered as the residue of a pole in one of those currents contracted with the remaining external polarization. The expansion keeps all color tensors explicit, allowing the color-factor shift to be applied term by term and cancelled by direct algebra.

Load-bearing premise

The proof assumes that the color-factor shift acts on every structure constant and generator exactly by the local momentum-difference rule given in the paper; if that rule fails to be compatible with the color algebra, the cancellation does not go through.

What would settle it

Explicitly compute any five- or six-point amplitude with two or more fundamental matter pairs plus gluons, apply the stated color-factor shift to its color factors, and check whether the numerical value of the amplitude remains exactly zero within machine precision; a nonzero residual would refute the claim.

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

If this is right

  • The fundamental BCJ relations for all tree amplitudes with fundamental matter follow at once from color-factor symmetry without a separate assumption of color-kinematic duality.
  • The same recursive cancellation supplies an all-multiplicity proof that had previously been obtained only by radiation-vertex expansion or by BCFW recursion.
  • Color-factor symmetry remains a gauge-invariant statement even when the kinematic numerators themselves do not obey Jacobi identities in every generalized gauge.
  • Any future extension of the perturbiner method to loops can be tested first by asking whether the same color-factor shift continues to leave the integrand invariant.

Where Pith is reading between the lines

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

  • The recursive structure that cancels the shift at tree level suggests a natural candidate for a loop-level color-factor symmetry once the appropriate off-shell currents are defined.
  • Because the proof never uses the explicit form of the matter wave-functions beyond the Dirac or Klein-Gordon equation, the same argument is expected to apply to any representation whose generators satisfy the fundamental commutation relations.
  • The cancellation pattern may offer a practical algorithm for constructing BCJ-satisfying numerators for mixed gluon-matter amplitudes without trial-and-error gauge fixing.

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

0 major / 4 minor

Summary. The paper proves that every tree-level amplitude of Yang-Mills theory coupled to an arbitrary number of fundamental scalars or Dirac spinors, provided it contains at least one external gluon, is invariant under the color-factor shift associated with that gluon. Starting from the classical equations of motion rewritten with auxiliary fields that keep them quadratic, the authors construct the color-dressed perturbiner expansion (Berends-Giele currents) and obtain a compact expression for the n-point amplitude (Eq. 4.37). They then apply the local color-factor shift (Eq. 3.2) and demonstrate, by exhaustive term-by-term cancellation using the recursion relations and the fundamental commutation relations, that the variation vanishes identically (Eq. 5.33). The resulting invariance immediately yields the fundamental BCJ relations for the Melia-Johansson-Ochirov partial amplitudes. An appendix recovers the known four-point scalar and spinor numerators and verifies the kinematic Jacobi identity.

Significance. The result supplies a transparent, recursive proof of color-factor symmetry for the full class of tree-level YM+matter amplitudes, thereby re-deriving the BCJ relations without assuming color-kinematic duality. The method extends the pure-YM perturbiner argument of ref. [19] in a uniform way and makes the cancellation mechanism fully explicit. Because the proof is all-orders in multiplicity and relies only on the classical equations of motion and standard color algebra, it strengthens the conceptual foundation of color-factor symmetry and offers a practical template for possible loop-level generalizations. The four-point checks confirm consistency with existing literature.

minor comments (4)
  1. In the pure-YM contribution S^\mu_YM (Eq. 5.6) the authors note a sign change relative to ref. [19] arising from the opposite plane-wave convention; a brief parenthetical reminder of that convention at the first appearance of the perturbiner ansatz would help readers who consult both papers.
  2. The lengthy intermediate expressions for S_i (Eqs. 5.8–5.18) are correct but dense; a short roadmap sentence at the beginning of Sec. 5 listing the four groups (S_(1), S_(2,1), S_(2,2), S_(3)) and the identities used to cancel each would improve readability.
  3. Appendix A recovers the literature numerators after setting the scalar wave-function factors to unity; an explicit statement that the overall phase and normalization conventions match those of refs. [9,10] would eliminate any residual ambiguity.
  4. A few typographical inconsistencies appear (e.g., “Yang-Mill” for “Yang-Mills” near Eq. 5.6, occasional missing spaces around · products). These are easily corrected in production.

Circularity Check

0 steps flagged

No significant circularity: the invariance under color-factor shifts is obtained by direct recursive cancellation from the classical equations of motion.

full rationale

The derivation begins from the Yang-Mills-plus-matter Lagrangian, rewrites the Euler-Lagrange equations with auxiliary fields so that they remain at most quadratic, constructs the color-dressed perturbiner expansion, and obtains the tree amplitude as the residue of the Berends-Giele current (4.36)–(4.37). The color-factor shift (3.2) is then applied term-by-term; the resulting expression S^μ is reduced by substituting the recursive relations (4.17)–(4.26) and the auxiliary-field identities (4.28)–(4.35). After regrouping, every pure-YM piece is proportional to k_P^μ (imported from the earlier pure-YM calculation) while the new matter contributions S_{(1)}, S_{(2,1)}, S_{(2,2)} and S_{(3)} cancel separately by momentum conservation, the Lorenz gauge, and the fundamental commutation relations. The final contraction with the external polarization vanishes, yielding δ_n A_n = 0. No parameter is fitted, no uniqueness theorem is imported from the authors’ prior work, and the definition of the shift is stated independently of the claim being proved. The four-point checks in Appendix A simply recover known numerators and serve as consistency tests. The argument is therefore self-contained algebraic verification rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is a pure derivation inside classical Yang-Mills theory. It inherits the standard Lagrangian, the color algebra, and the definition of the color-factor shift from prior literature; it introduces no free parameters and no new dynamical entities.

axioms (3)
  • domain assumption Yang-Mills Lagrangian coupled to fundamental scalars and Dirac fermions (eq. 4.1) together with the Lorenz gauge condition.
    All equations of motion and the perturbiner expansion are derived from this Lagrangian; the result is theory-specific.
  • standard math Color algebra [T^a,T^b]=f^{abc}T^c and the associated Jacobi identity for structure constants.
    Used repeatedly to cancel terms after the color-factor shift is applied (e.g., eqs. 5.26–5.31).
  • domain assumption Definition of the color-factor shift δ_a on generators and structure constants (eq. 3.2).
    Taken from earlier color-factor-symmetry papers; the entire cancellation proof is relative to this definition.

pith-pipeline@v1.1.0-grok45 · 30087 in / 2172 out tokens · 20172 ms · 2026-07-14T01:04:28.276758+00:00 · methodology

0 comments
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.

discussion (0)

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

Reference graph

Works this paper leans on

53 extracted references · 47 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]

    Perturbiner methods in scattering amplitude

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

    Pith/arXiv arXiv

  42. [42]

    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

  43. [43]

    The Color Structure of Gluon Emission

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

    1988

  44. [44]

    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

  45. [45]

    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

  46. [46]

    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

  47. [47]

    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

  48. [48]

    Dyck words and multiquark primitive amplitudes

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

    Pith/arXiv arXiv 2013

  49. [49]

    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

  50. [50]

    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

  51. [51]

    Covariant color-kinematics duality

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

    Pith/arXiv arXiv 2021

  52. [52]

    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

  53. [53]

    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)]. 29

    Pith/arXiv arXiv 2010