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arxiv: 2305.04620 · v1 · submitted 2023-05-08 · ✦ hep-th

Tree and 1-loop fundamental BCJ relations from soft theorems

Fang-Stars Wei , Kang Zhou This is my paper

Pith reviewed 2026-05-24 09:00 UTC · model grok-4.3

classification ✦ hep-th
keywords BCJ relationssoft theoremsbi-adjoint scalar theorycolor-ordered amplitudesone-loop integrandsAdler zeronon-linear sigma modelBorn-Infeld theory
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0 comments X

The pith

The leading soft theorem for scalars derives the fundamental BCJ relation among double color-ordered tree amplitudes in bi-adjoint scalar theory and extends it to one-loop integrands.

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

The paper derives the fundamental BCJ relation for double color-ordered tree amplitudes of bi-adjoint scalar theory directly from the leading soft theorem applied to external scalars. This replaces previous derivations with a new approach based on soft limits. The same relation is then shown to hold for one-loop Feynman integrands. The method is also applied to recover the Adler zero condition for tree amplitudes in the non-linear sigma model and Born-Infeld theory. A reader would care because these relations constrain how amplitudes factor and simplify calculations across scalar, gauge, and gravity theories.

Core claim

The paper establishes that the leading soft theorem for external scalars in bi-adjoint scalar theory directly produces the fundamental BCJ relation among double color-ordered tree amplitudes. The same soft-limit argument is used to obtain the analogous relation at the level of one-loop Feynman integrands. The relation is further employed to reproduce the Adler zero for the tree amplitudes of the non-linear sigma model and Born-Infeld theories.

What carries the argument

The leading soft theorem for external scalars, which fixes the singular behavior of color-ordered amplitudes when one external momentum approaches zero and thereby enforces the linear dependence expressed by the fundamental BCJ relation.

If this is right

  • The fundamental BCJ relation among double color-ordered amplitudes follows immediately from the soft theorem at tree level.
  • The identical relation applies to the integrands of one-loop Feynman diagrams in the same theory.
  • The relation accounts for the vanishing of amplitudes in the soft limit (Adler zero) for the non-linear sigma model and Born-Infeld theory.

Where Pith is reading between the lines

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

  • The soft-theorem method may supply a uniform way to obtain BCJ-type identities in other scalar theories that possess a well-defined soft limit.
  • If the one-loop generalization survives integration, it would constrain the structure of loop-level color-kinematics relations without needing explicit integrand construction.

Load-bearing premise

The leading soft theorem for external scalars holds exactly for the color-ordered amplitudes of bi-adjoint scalar theory with no additional corrections from the theory's interactions.

What would settle it

An explicit computation of a double color-ordered tree amplitude in which the soft limit of one leg fails to reproduce the expected linear combination that defines the fundamental BCJ relation.

Figures

Figures reproduced from arXiv: 2305.04620 by Fang-Stars Wei, Kang Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1: Two 5-point diagrams [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Diagram for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The overall sign + under the new convention. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Decomposition of 1-loop Feynman integrand. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

We provide a new derivation of the fundamental BCJ relation among double color ordered tree amplitudes of bi-adjoint scalar theory, based on the leading soft theorem for external scalars. Then, we generalize the fundamental BCJ relation to $1$-loop Feynman integrands. We also use the fundamental BCJ relation to understand the Adler's zero for tree amplitudes of non-linear Sigma model and Born-Infeld theories.

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 / 2 minor

Summary. The paper claims to provide a new derivation of the fundamental BCJ relation among double color-ordered tree amplitudes of bi-adjoint scalar theory based on the leading soft theorem for external scalars. It then generalizes the fundamental BCJ relation to 1-loop Feynman integrands and uses the relation to understand Adler's zero for tree amplitudes of non-linear sigma model and Born-Infeld theories.

Significance. If the central derivation holds without circularity, the work is significant for offering an alternative route to BCJ relations via soft theorems rather than direct color-kinematics arguments, with the 1-loop generalization and Adler-zero applications providing additional utility in amplitude relations. No machine-checked proofs or parameter-free derivations are highlighted, but the direct use of soft theorems is a clear strength if substantiated.

major comments (2)
  1. [tree-level derivation (section deriving the fundamental BCJ from soft theorem)] The tree-level derivation applies the leading soft theorem (scalar soft factor proportional to sum_i (epsilon · k_i)/(k_soft · k_i)) directly to the double color-ordered partial amplitudes. The manuscript must explicitly demonstrate that this application to the ordered sector does not presuppose the BCJ relation via the color decomposition commuting with the soft limit; this is load-bearing for the claim of a new derivation.
  2. [1-loop section] For the 1-loop generalization, the extension of the soft theorem to Feynman integrands requires clarification on whether the soft factor acts before or after loop integration and any assumptions on the loop momentum; without this, the generalization claim rests on an unstated step.
minor comments (2)
  1. Notation for the double color-ordered amplitudes could be introduced more explicitly at the start of the tree-level section for clarity.
  2. The abstract mentions the applications to NLSM and BI but does not specify which form of the BCJ relation is used; a brief parenthetical would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points that require additional clarification. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation of the derivations.

read point-by-point responses

  1. Referee: [tree-level derivation (section deriving the fundamental BCJ from soft theorem)] The tree-level derivation applies the leading soft theorem (scalar soft factor proportional to sum_i (epsilon · k_i)/(k_soft · k_i)) directly to the double color-ordered partial amplitudes. The manuscript must explicitly demonstrate that this application to the ordered sector does not presuppose the BCJ relation via the color decomposition commuting with the soft limit; this is load-bearing for the claim of a new derivation.

    Authors: We agree that an explicit demonstration is necessary to rule out circularity. The leading soft theorem is first applied to the complete bi-adjoint scalar amplitude (whose soft behavior follows from the Feynman rules without reference to BCJ). The double color-ordered amplitudes are subsequently isolated via the standard color decomposition of the bi-adjoint theory. Because the soft factor itself is independent of color ordering, the same soft limit can be taken on each ordered sector separately; equating the two expressions then yields the fundamental BCJ relation. No prior use of BCJ is required for the color decomposition to commute with the soft limit. We have added a dedicated paragraph in the tree-level section that walks through this ordering of steps and explicitly notes that the soft factor is color-blind. revision: yes

  2. Referee: [1-loop section] For the 1-loop generalization, the extension of the soft theorem to Feynman integrands requires clarification on whether the soft factor acts before or after loop integration and any assumptions on the loop momentum; without this, the generalization claim rests on an unstated step.

    Authors: We thank the referee for noting this ambiguity. The soft factor is applied directly to the 1-loop Feynman integrand (prior to any loop integration), treating the loop momentum as an independent off-shell variable that does not participate in the soft limit. This is the standard integrand-level formulation used in the soft-theorem literature. We have revised the 1-loop section to state these conventions explicitly, including a sentence confirming that the soft theorem is imposed at the integrand level with no additional assumptions on the loop momentum. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation self-contained

full rationale

The paper derives the fundamental BCJ relation for tree amplitudes from the leading soft theorem applied to color-ordered amplitudes in bi-adjoint scalar theory, then extends to 1-loop integrands. No equations, sections, or self-citations are exhibited in the provided text that reduce the claimed result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The soft theorem is invoked as an independent external input whose validity is assumed to hold exactly, with no visible reduction of the BCJ output to the input by construction. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities.

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Forward citations

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · cited by 2 Pith papers · 57 internal anchors

  1. [1]

    Applying this sewing manipulation, one can generalize the tree level fundamental BCJ relation to the 1-loop level, as will be seen in subsection.IV C. B. Forward limit method The 1-loop Feynman integrands can be generated from the corresponding tree amplitudes, via the so called forward limit procedure. For instance, the 1-loop CHY formulas can be obtaine...

    work page

  2. [2]

    Bremsstrahlung of very low-energy quanta in elementary particle collisions,

    F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,” Phys. Rev. 110, 974 (1958)

    work page 1958

  3. [3]

    Infrared photons and gravitons,

    S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965)

    work page 1965

  4. [4]

    Evidence for a New Soft Graviton Theorem

    F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Soft sub-leading divergences in Yang-Mills amplitudes

    E. Casali, “Soft sub-leading divergences in Yang-Mills amplitudes,” JHEP 08, 077 (2014) doi:10.1007/JHEP08(2014)077 [arXiv:1404.5551 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2014)077 2014

  6. [6]

    New Recursion Relations for Tree Amplitudes of Gluons

    R. Britto, F. Cachazo and B. Feng, “New recursion relations for tree amplitudes of gluons,” Nucl. Phys. B715, 499-522 (2005) doi:10.1016/j.nuclphysb.2005.02.030 [arXiv:hep-th/0412308 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2005.02.030 2005

  7. [7]

    Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory

    R. Britto, F. Cachazo, B. Feng and E. Witten, “Direct proof of tree-level recursion relation in Yang-Mills theory,” Phys. Rev. Lett. 94, 181602 (2005) doi:10.1103/PhysRevLett.94.181602 [arXiv:hep-th/0501052 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.94.181602 2005

  8. [8]

    Subleading soft theorem in arbitrary dimension from scattering equations

    B. U. W. Schwab and A. Volovich, “Subleading Soft Theorem in Arbitrary Di- mensions from Scattering Equations,” Phys. Rev. Lett. 113, no.10, 101601 (2014) doi:10.1103/PhysRevLett.113.101601 [arXiv:1404.7749 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.113.101601 2014

  9. [9]

    Soft Graviton Theorem in Arbitrary Dimensions

    N. Afkhami-Jeddi, “Soft Graviton Theorem in Arbitrary Dimensions,” [arXiv:1405.3533 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Scattering Equations and KLT Orthogonality

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Kawai-Lewellen-Tye Orthog- onality,” Phys. Rev. D90 (2014) no. 6, 065001, arXiv:1306.6575 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  11. [11]

    Scattering of Massless Particles in Arbitrary Dimension

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett. 113 (2014) no. 17, 171601, arXiv:1307.2199 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  12. [12]

    Scattering of Massless Particles: Scalars, Gluons and Gravitons

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,” JHEP 1407 (2014) 033, arXiv:1309.0885 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  13. [13]

    Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations

    F. Cachazo, S. He and E. Y. Yuan, “Einstein-Yang-Mills Scattering Amplitudes From Scat- tering Equations,” JHEP 1501, 121 (2015) [arXiv:1409.8256 [hep-th]]. 20

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  14. [14]

    Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM

    F. Cachazo, S. He and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP 1507, 149 (2015) [arXiv:1412.3479 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  15. [15]

    On BMS Invariance of Gravitational Scattering

    A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07, 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2014)152 2014

  16. [16]

    Asymptotic Symmetries of Yang-Mills Theory

    A. Strominger, JHEP 07, 151 (2014) doi:10.1007/JHEP07(2014)151 [arXiv:1308.0589 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2014)151 2014

  17. [17]

    T. He, V. Lysov, P. Mitra and A. Strominger, JHEP 05, 151 (2015) doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2015)151 2015

  18. [18]

    Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, JHEP 08, 058 (2014) doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2014)058 2014

  19. [19]

    Gravitational Memory, BMS Supertranslations and Soft Theorems

    A. Strominger and A. Zhiboedov, JHEP 01, 086 (2016) doi:10.1007/JHEP01(2016)086 [arXiv:1411.5745 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2016)086 2016

  20. [20]

    New Gravitational Memories

    S. Pasterski, A. Strominger and A. Zhiboedov, JHEP 12, 053 (2016) doi:10.1007/JHEP12(2016)053 [arXiv:1502.06120 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2016)053 2016

  21. [21]

    Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited

    G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105, 111103 (2010) [arXiv:0909.2617 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  22. [22]

    Supertranslations call for superrotations

    G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG 2010, 010 (2010) [arXiv:1102.4632 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    BMS charge algebra

    G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 1112, 105 (2011) [arXiv:1106.0213 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  24. [24]

    On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons

    Z. Bern, S. Davies and J. Nohle, “On Loop Corrections to sub-leading Soft Behavior of Gluons and Gravitons,” arXiv:1405.1015 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  25. [25]

    Loop Corrections to Soft Theorems in Gauge Theories and Gravity

    S. He, Y. -t. Huang and C. Wen, “Loop Corrections to Soft Theorems in Gauge Theories and Gravity,” arXiv:1405.1410 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  26. [26]

    Are Soft Theorems Renormalized?

    F. Cachazo and E. Y. Yuan, “Are Soft Theorems Renormalized?,” arXiv:1405.3413 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    More on Soft Theorems: Trees, Loops and Strings

    M. Bianchi, S. He, Y. t. Huang and C. Wen, “More on Soft Theorems: Trees, Loops and Strings,” Phys. Rev. D 92, no.6, 065022 (2015) doi:10.1103/PhysRevD.92.065022 [arXiv:1406.5155 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.065022 2015

  28. [28]

    Asymptotic symmetries and subleading soft graviton theorem

    M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D 90, no.12, 124028 (2014) doi:10.1103/PhysRevD.90.124028 [arXiv:1408.2228 [hep-th]]. 21

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.124028 2014

  29. [29]

    Sub-subleading soft gravitons and large diffeomorphisms

    M. Campiglia and A. Laddha, “Sub-subleading soft gravitons and large diffeomorphisms,” JHEP 01, 036 (2017) doi:10.1007/JHEP01(2017)036 [arXiv:1608.00685 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2017)036 2017

  30. [30]

    Soft Photon and Graviton Theorems in Effective Field Theory

    H. Elvang, C. R. T. Jones and S. G. Naculich, “Soft Photon and Graviton The- orems in Effective Field Theory,” Phys. Rev. Lett. 118, no.23, 231601 (2017) doi:10.1103/PhysRevLett.118.231601 [arXiv:1611.07534 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.118.231601 2017

  31. [31]

    On the exactness of soft theorems

    A. L. Guerrieri, Y. t. Huang, Z. Li and C. Wen, “On the exactness of soft theorems,” JHEP 12, 052 (2017) doi:10.1007/JHEP12(2017)052 [arXiv:1705.10078 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2017)052 2017

  32. [32]

    Soft pion theorem, asymptotic symmetry and new memory effect

    Y. Hamada and S. Sugishita, “Soft pion theorem, asymptotic symmetry and new memory effect,” JHEP 11, 203 (2017) doi:10.1007/JHEP11(2017)203 [arXiv:1709.05018 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep11(2017)203 2017

  33. [33]

    Note on asymptotic symmetries and soft gluon theorems,

    P. Mao and J. B. Wu, “Note on asymptotic symmetries and soft gluon theorems,” Phys. Rev. D 96, no.6, 065023 (2017) doi:10.1103/PhysRevD.96.065023 [arXiv:1704.05740 [hep-th]]

    work page doi:10.1103/physrevd.96.065023 2017

  34. [34]

    On the Symmetry Foundation of Double Soft Theorems

    Z. z. Li, H. h. Lin and S. q. Zhang, “On the Symmetry Foundation of Double Soft Theorems,” JHEP 12, 032 (2017) doi:10.1007/JHEP12(2017)032 [arXiv:1710.00480 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2017)032 2017

  35. [35]

    The B-field soft theorem and its unification with the graviton and dilaton

    P. Di Vecchia, R. Marotta and M. Mojaza, “The B-field soft theorem and its unifica- tion with the graviton and dilaton,” JHEP 10, 017 (2017) doi:10.1007/JHEP10(2017)017 [arXiv:1706.02961 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2017)017 2017

  36. [36]

    Exploring soft constraints on effective actions

    M. Bianchi, A. L. Guerrieri, Y. t. Huang, C. J. Lee and C. Wen, “Exploring soft constraints on effective actions,” JHEP 10, 036 (2016) doi:10.1007/JHEP10(2016)036 [arXiv:1605.08697 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2016)036 2016

  37. [37]

    Subleading Soft Theorem for Multiple Soft Gravitons

    S. Chakrabarti, S. P. Kashyap, B. Sahoo, A. Sen and M. Verma, “Subleading Soft The- orem for Multiple Soft Gravitons,” JHEP 12, 150 (2017) doi:10.1007/JHEP12(2017)150 [arXiv:1707.06803 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2017)150 2017

  38. [38]

    Subleading Soft Graviton Theorem for Loop Amplitudes

    A. Sen, “Subleading Soft Graviton Theorem for Loop Amplitudes,” JHEP 11, 123 (2017) doi:10.1007/JHEP11(2017)123 [arXiv:1703.00024 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep11(2017)123 2017

  39. [39]

    Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities

    Y. Hamada and G. Shiu, “Infinite Set of Soft Theorems in Gauge-Gravity The- ories as Ward-Takahashi Identities,” Phys. Rev. Lett. 120, no.20, 201601 (2018) doi:10.1103/PhysRevLett.120.201601 [arXiv:1801.05528 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.120.201601 2018

  40. [40]

    Effective Field Theories from Soft Limits

    C. Cheung, K. Kampf, J. Novotny and J. Trnka, “Effective Field Theories from Soft Limits of Scattering Amplitudes,” Phys. Rev. Lett. 114, no.22, 221602 (2015) doi:10.1103/PhysRevLett.114.221602 [arXiv:1412.4095 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.114.221602 2015

  41. [41]

    Recursion relations from soft theorems

    H. Luo and C. Wen, “Recursion relations from soft theorems,” JHEP 03, 088 (2016) 22 doi:10.1007/JHEP03(2016)088 [arXiv:1512.06801 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2016)088 2016

  42. [42]

    Soft Bootstrap and Supersymmetry

    H. Elvang, M. Hadjiantonis, C. R. T. Jones and S. Paranjape, “Soft Bootstrap and Super- symmetry,” JHEP 01, 195 (2019) doi:10.1007/JHEP01(2019)195 [arXiv:1806.06079 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2019)195 2019

  43. [43]

    Extensions of Theories from Soft Limits

    F. Cachazo, P. Cha and S. Mizera, “Extensions of Theories from Soft Limits,” JHEP 06, 170 (2016) doi:10.1007/JHEP06(2016)170 [arXiv:1604.03893 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep06(2016)170 2016

  44. [44]

    Scattering Amplitudes from Soft Theorems and Infrared Behavior

    L. Rodina, “Scattering Amplitudes from Soft Theorems and Infrared Behavior,” Phys. Rev. Lett. 122, no.7, 071601 (2019) doi:10.1103/PhysRevLett.122.071601 [arXiv:1807.09738 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.122.071601 2019

  45. [45]

    Constructing Amplitudes from Their Soft Limits

    C. Boucher-Veronneau and A. J. Larkoski, “Constructing Amplitudes from Their Soft Limits,” JHEP 09, 130 (2011) doi:10.1007/JHEP09(2011)130 [arXiv:1108.5385 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2011)130 2011

  46. [46]

    The Tree Formula for MHV Graviton Amplitudes

    D. Nguyen, M. Spradlin, A. Volovich and C. Wen, “The Tree Formula for MHV Graviton Amplitudes,” JHEP 07, 045 (2010) doi:10.1007/JHEP07(2010)045 [arXiv:0907.2276 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2010)045 2010

  47. [47]

    Tree level amplitudes from soft theorems

    K. Zhou, JHEP 03, 021 (2023) doi:10.1007/JHEP03(2023)021 [arXiv:2212.12892 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2023)021 2023

  48. [48]

    A Relation Between Tree Amplitudes of Closed and Open Strings,

    H. Kawai, D. C. Lewellen and S. H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl. Phys. B 269, 1 (1986)

    work page 1986

  49. [49]

    New Relations for Gauge-Theory Amplitudes

    Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys. Rev. D 78, 085011 (2008) [arXiv:0805.3993 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  50. [50]

    Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity

    M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, “Scattering amplitudes in N = 2 Maxwell-Einstein and Yang-Mills/Einstein supergravity,” JHEP 1501, 081 (2015) doi:10.1007/JHEP01(2015)081 [arXiv:1408.0764 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2015)081 2015

  51. [51]

    Color-Kinematics Duality for QCD Amplitudes

    H. Johansson and A. Ochirov, “Color-Kinematics Duality for QCD Amplitudes,” JHEP 1601, 170 (2016) doi:10.1007/JHEP01(2016)170 [arXiv:1507.00332 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2016)170 2016

  52. [52]

    Double copy for massive quantum particles with spin,

    H. Johansson and A. Ochirov, “Double copy for massive quantum particles with spin,” JHEP 1909, 040 (2019) doi:10.1007/JHEP09(2019)040 [arXiv:1906.12292 [hep-th]]

    work page doi:10.1007/jhep09(2019)040 1909

  53. [53]

    Expansion of Einstein-Yang-Mills Amplitude

    C. H. Fu, Y. J. Du, R. Huang and B. Feng, “Expansion of Einstein-Yang-Mills Amplitude,” JHEP 1709, 021 (2017) doi:10.1007/JHEP09(2017)021 [arXiv:1702.08158 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2017)021 2017

  54. [54]

    Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame

    F. Teng and B. Feng, “Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame,” JHEP 1705, 075 (2017) [arXiv:1703.01269 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [55]

    BCJ numerators from reduced Pfaffian

    Y. J. Du and F. Teng, “BCJ numerators from reduced Pfaffian,” JHEP 1704, 033 (2017) [arXiv:1703.05717 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  56. [56]

    Expansion of All Multitrace Tree Level EYM Amplitudes

    Y. J. Du, B. Feng and F. Teng, “Expansion of All Multitrace Tree Level EYM Amplitudes,” 23 JHEP 1712, 038 (2017) [arXiv:1708.04514 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  57. [57]

    Expansion of EYM theory by Differential Operators,

    B. Feng, X. Li and K. Zhou, “Expansion of EYM theory by Differential Operators,” arXiv:1904.05997 [hep-th]

    work page arXiv 1904

  58. [58]

    Unified web for expansions of amplitudes,

    K. Zhou, “Unified web for expansions of amplitudes,” JHEP 10, 195 (2019) doi:10.1007/JHEP10(2019)195 [arXiv:1908.10272 [hep-th]]

    work page doi:10.1007/jhep10(2019)195 2019

  59. [59]

    On Primary Relations at Tree-level in String Theory and Field Theory

    Q. Ma, Y. J. Du and Y. X. Chen, “On Primary Relations at Tree-level in String Theory and Field Theory,” JHEP 02, 061 (2012) doi:10.1007/JHEP02(2012)061 [arXiv:1109.0685 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2012)061 2012

  60. [60]

    One-loop Scattering Equations and Amplitudes from Forward Limit

    S. He and E. Y. Yuan, “One-loop Scattering Equations and Amplitudes from Forward Limit,” Phys. Rev. D 92, no.10, 105004 (2015) doi:10.1103/PhysRevD.92.105004 [arXiv:1508.06027 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.105004 2015

  61. [61]

    One-Loop Corrections from Higher Dimensional Tree Amplitudes

    F. Cachazo, S. He and E. Y. Yuan, “One-Loop Corrections from Higher Dimensional Tree Amplitudes,” JHEP 08, 008 (2016) doi:10.1007/JHEP08(2016)008 [arXiv:1512.05001 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2016)008 2016

  62. [62]

    CHY-construction of Planar Loop Integrands of Cubic Scalar Theory

    B. Feng, “CHY-construction of Planar Loop Integrands of Cubic Scalar Theory,” JHEP 05, 061 (2016) doi:10.1007/JHEP05(2016)061 [arXiv:1601.05864 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2016)061 2016

  63. [63]

    One-loop CHY-Integrand of Bi-adjoint Scalar Theory,

    B. Feng and C. Hu, “One-loop CHY-Integrand of Bi-adjoint Scalar Theory,” JHEP 02, 187 (2020) doi:10.1007/JHEP02(2020)187 [arXiv:1912.12960 [hep-th]]

    work page doi:10.1007/jhep02(2020)187 2020

  64. [64]

    New Representations of the Perturbative S-Matrix

    C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, S. Caron-Huot, P. H. Damgaard and B. Feng, “New Representations of the Perturbative S-Matrix,” Phys. Rev. Lett. 116, no.6, 061601 (2016) doi:10.1103/PhysRevLett.116.061601 [arXiv:1509.02169 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.116.061601 2016

  65. [65]

    Progress in One-Loop QCD Computations

    Z. Bern, L. J. Dixon and D. A. Kosower, “Progress in one loop QCD computations,” Ann. Rev. Nucl. Part. Sci. 46, 109-148 (1996) doi:10.1146/annurev.nucl.46.1.109 [arXiv:hep-ph/9602280 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1146/annurev.nucl.46.1.109 1996

  66. [66]

    MULTI - GLUON CROSS-SECTIONS AND FIVE JET PRODUC- TION AT HADRON COLLIDERS,

    R. Kleiss and H. Kuijf, “MULTI - GLUON CROSS-SECTIONS AND FIVE JET PRODUC- TION AT HADRON COLLIDERS,” Nucl. Phys. B 312, 616 (1989)

    work page 1989

  67. [67]

    On General BCJ Relation at One-loop Level in Yang-Mills Theory

    Y. J. Du and H. Luo, “On General BCJ Relation at One-loop Level in Yang-Mills Theory,” JHEP 01, 129 (2013) doi:10.1007/JHEP01(2013)129 [arXiv:1207.4549 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2013)129 2013

  68. [68]

    Zhou and F

    K. Zhou and F. Wei, in prepaireation. 24

    work page