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arxiv: 2502.07173 · v1 · submitted 2025-02-11 · ✦ hep-th

Note on hidden zeros and expansions of tree-level amplitudes

Hao Huang , Ye Yang , Kang Zhou This is my paper

Pith reviewed 2026-05-23 04:22 UTC · model grok-4.3

classification ✦ hep-th
keywords hidden zerostree-level amplitudesbi-adjoint scalarYang-Millsgravityuniversal expansionsscattering amplitudeskinematic zeros
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The pith

Hidden zeros in tree-level amplitudes of Yang-Mills, gravity and related theories arise from zeros of bi-adjoint scalar amplitudes via universal expansions.

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

The paper establishes that hidden zeros appearing in tree-level scattering amplitudes of Yang-Mills, non-linear sigma model, special Galileon, Dirac-Born-Infeld, and gravity theories follow directly from zeros of bi-adjoint scalar amplitudes. It applies universal expansions that rewrite the amplitudes of these theories as combinations of bi-adjoint scalar amplitudes. Because the scalar zeros are simple to prove, the zeros in the other theories become immediate consequences. For gravitational amplitudes the same kinematic conditions that produce zeros can generate apparent propagator divergences, and the expansions supply the explicit cancellation mechanism.

Core claim

Universal expansions that map amplitudes of Yang-Mills, NLSM, sGal, DBI and gravity onto bi-adjoint scalar amplitudes show that the hidden zeros of the former are inherited from zeros of the latter. The scalar zeros are established by direct inspection of the amplitude expressions. For unordered amplitudes such as those of gravity, the kinematic conditions inducing zeros introduce potential divergences in propagators; the expansions make the cancellation of these divergences manifest through the structure of the summed terms.

What carries the argument

Universal expansions of tree-level amplitudes that expand amplitudes of different theories to those of bi-adjoint scalar theory.

If this is right

  • Zeros of all listed amplitudes are proved once the bi-adjoint scalar zeros are known.
  • Any kinematic configuration satisfying the zero condition forces the expanded amplitudes to vanish.
  • For gravity the expansions guarantee that apparent propagator divergences cancel exactly at the zero loci.
  • A single proof strategy covers hidden zeros across Yang-Mills, gravity, and the other theories.

Where Pith is reading between the lines

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

  • Similar expansions, if available at loop level, could locate hidden zeros in higher-order amplitudes.
  • Numerical verification of amplitude vanishing can be reduced to checking the simpler scalar case.
  • The expansions may connect hidden zeros to other known amplitude identities such as color-kinematics relations.
  • Specific low-point kinematic limits could be used to test whether additional hidden structures appear.

Load-bearing premise

The universal expansions are valid and complete for the kinematic configurations that produce the hidden zeros.

What would settle it

An explicit kinematic point where the bi-adjoint scalar amplitude vanishes but the corresponding Yang-Mills or gravitational amplitude remains non-zero, while the expansion is still applied.

read the original abstract

In this note, we derive and interpret hidden zeros of tree-level amplitudes of various theories, including Yang-Mills, non-linear sigma model, special Galileon, Dirac-Born-Infeld, and gravity, by utilizing universal expansions of tree-level amplitudes which expand amplitudes of different theories to those of bi-adjoint scalar theory. Hidden zeros of above amplitudes are attributed to zeros of bi-adjoint scalar amplitudes which can be easily proved. For unordered amplitudes such as gravitational ones, the kinematic condition for zeros causes potential divergences of propagators. We also show the detailed mechanism of eliminating such divergences.

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 / 0 minor

Summary. The manuscript claims that hidden zeros in tree-level amplitudes of Yang-Mills, non-linear sigma model, special Galileon, Dirac-Born-Infeld, and gravity theories follow directly from the zeros of bi-adjoint scalar (BAS) amplitudes via universal expansions that relate the different theories to BAS. It attributes the zeros to the easily proved BAS zeros and, for unordered amplitudes such as gravity, provides a mechanism showing how potential propagator divergences cancel under the relevant kinematic conditions.

Significance. If the expansions remain valid in the singular kinematic regime, the result supplies a unified and economical account of hidden zeros across several theories by reduction to the BAS case, without new parameters. The explicit treatment of divergence cancellation for gravitational amplitudes addresses a technical point that could otherwise obstruct application to unordered amplitudes. This strengthens the case for using BAS as a reference theory for amplitude properties.

major comments (2)
  1. The central claim requires that the universal expansions continue to hold when the chosen kinematic conditions render certain propagators on-shell (singular). The abstract states that divergences cancel, but this cancellation occurs after the expansion step and does not by itself establish that the expansion formulae themselves remain complete and valid in the singular limit. A concrete derivation or explicit check of the expansion under these conditions is needed to close the gap.
  2. The argument rests on the completeness of the cited universal expansions for the specific kinematic configurations that induce the zeros. If any derivation of those expansions assumed off-shell propagators or analytic continuation away from the mass shell, the reduction to BAS zeros would not automatically apply. The manuscript should identify the precise location where this applicability is verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which help clarify the scope of our claims. Below we respond point by point to the two major remarks.

read point-by-point responses

  1. Referee: The central claim requires that the universal expansions continue to hold when the chosen kinematic conditions render certain propagators on-shell (singular). The abstract states that divergences cancel, but this cancellation occurs after the expansion step and does not by itself establish that the expansion formulae themselves remain complete and valid in the singular limit. A concrete derivation or explicit check of the expansion under these conditions is needed to close the gap.

    Authors: We agree that an explicit verification of the expansions inside the singular kinematic region strengthens the argument. The expansions themselves are algebraic identities obtained from the same set of Feynman rules that define the amplitudes; they therefore remain valid when a subset of propagators go on-shell. Nevertheless, to address the referee’s concern directly we will add a short subsection that performs the expansion explicitly for one representative theory (Yang-Mills) under the precise kinematic conditions that produce the hidden zero, confirming that no extra singular terms appear before the cancellation mechanism is applied. revision: yes

  2. Referee: The argument rests on the completeness of the cited universal expansions for the specific kinematic configurations that induce the zeros. If any derivation of those expansions assumed off-shell propagators or analytic continuation away from the mass shell, the reduction to BAS zeros would not automatically apply. The manuscript should identify the precise location where this applicability is verified.

    Authors: The cited universal expansions are derived directly for on-shell tree amplitudes and do not invoke off-shell assumptions; their completeness follows from the same diagrammatic counting used for the ordinary amplitudes. The manuscript already applies them to the zero-inducing kinematics in Sections 3 and 4 and demonstrates consistency via the gravity cancellation. To make the verification explicit we will insert a clarifying sentence that points to the relevant equations and states that the expansions hold in the singular limit because they are identities among rational functions of the kinematic invariants. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent BAS zero proof

full rationale

The paper's chain uses cited universal expansions to map other amplitudes onto bi-adjoint scalar (BAS) amplitudes, then attributes the hidden zeros to BAS zeros that are described as independently and easily provable. The text also supplies an explicit mechanism for canceling propagator divergences under the chosen kinematics. No quoted step reduces a central prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the final attribution step stands on the separate BAS result rather than on the expansions alone. This is the normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and completeness of the universal expansions to BAS amplitudes; these are treated as given background rather than re-derived here.

axioms (1)
  • domain assumption Universal expansions exist that express tree-level amplitudes of Yang-Mills, NLSM, sGal, DBI, and gravity in terms of bi-adjoint scalar amplitudes.
    Invoked in the first sentence of the abstract as the tool used to attribute the zeros.

pith-pipeline@v0.9.0 · 5614 in / 1238 out tokens · 35442 ms · 2026-05-23T04:22:52.260223+00:00 · methodology

discussion (0)

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Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Universal Interpretation of Hidden Zero and $2$-Split of Tree-Level Amplitudes Using Feynman Diagrams, Part $\mathbf{I}$: ${\rm Tr}(\phi^3)$, NLSM and YM

    hep-th 2026-04 unverdicted novelty 6.0

    A universal diagrammatic interpretation unifies hidden zeros (from massless on-shell conditions) and 2-splits (from double-line separation) in Tr(φ³), NLSM, and YM tree amplitudes using extended shuffle factorization ...

  2. Towards New Hidden Zero and $2$-Split of Loop-Level Feynman Integrands in ${\rm Tr}(\phi^3)$ Model

    hep-th 2026-04 unverdicted novelty 6.0

    Loop-level hidden zeros and 2-split structures are found in Tr(φ³) Feynman integrands with simple kinematic conditions, generalizing the tree-level case to an L-loop integrand expressed as a sum over L+1 terms each wi...

  3. Hidden zeros for higher-derivative YM and GR amplitudes at tree-level

    hep-th 2025-10 unverdicted novelty 6.0

    Hidden zeros extend to higher-derivative tree-level gluon and graviton amplitudes, with systematic cancellation of propagator singularities shown via bi-adjoint scalar expansions.

  4. A new recursion relation for tree-level NLSM amplitudes based on hidden zeros

    hep-th 2025-08 unverdicted novelty 6.0

    A recursion for NLSM tree amplitudes based on hidden zeros reproduces the Adler zero, generates amplitudes from Tr(φ³) via δ-shift, expands them into bi-adjoint scalars, and claims these plus factorization uniquely de...

  5. Can Locality, Unitarity, and Hidden Zeros Completely Determine Tree-Level Amplitudes?

    hep-th 2026-04 unverdicted novelty 5.0

    Locality, unitarity, and hidden zeros determine tree-level YM and NLSM amplitudes by reconstructing their soft theorems.

  6. $2$-split from Feynman diagrams and Expansions

    hep-th 2025-08 unverdicted novelty 5.0

    Proof via Feynman diagrams that tree-level BAS⊕X amplitudes with X=YM,NLSM,GR obey 2-split under kinematic conditions, extended to pure X amplitudes with byproduct universal expansions of X currents into BAS currents.

  7. Soft theorems of tree-level ${\rm Tr}(\phi^3)$, YM and NLSM amplitudes from $2$-splits

    hep-th 2025-05 unverdicted novelty 5.0

    Extends a 2-split factorization approach to reproduce known leading and sub-leading soft theorems for Tr(φ³) and YM single-soft and NLSM double-soft amplitudes while deriving higher-order universal forms and a kinemat...

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 7 Pith papers · 24 internal anchors

  1. [1]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  2. [2]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  3. [3]

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, The duality between color and kinematics and its applications , J. Phys. A 57 (2024), no. 33 333002, [ arXiv:1909.01358]

    work page arXiv 2024

  4. [4]

    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]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  5. [5]

    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 07 (2014) 033, [ arXiv:1309.0885]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  6. [6]

    Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations

    F. Cachazo, S. He, and E. Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations , JHEP 01 (2015) 121, [ arXiv:1409.8256]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  7. [7]

    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 07 (2015) 149, [ arXiv:1412.3479]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  8. [8]

    Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

    N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet , JHEP 05 (2018) 096, [ arXiv:1711.09102]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    All Loop Scattering as a Counting Problem,

    N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon, and H. Thomas, All Loop Scattering As A Counting Problem, arXiv:2309.15913

    work page arXiv

  10. [10]

    Arkani-Hamed, H

    N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon, and H. Thomas, All Loop Scattering For All Multiplicity, arXiv:2311.09284

    work page arXiv

  11. [11]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory , arXiv:2401.00041

    work page arXiv

  12. [12]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, Nonlinear Sigma model amplitudes to all loop orders are contained in the Tr( Φ3) theory, Phys. Rev. D 110 (2024), no. 6 065018, [ arXiv:2401.05483]

    work page arXiv 2024

  13. [13]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo, and S. He, Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons , JHEP 10 (2024) 231, [ arXiv:2312.16282]

    work page arXiv 2024

  14. [14]

    Rodina, Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes in Tr( ϕ3) theory, Phys

    L. Rodina, Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes in Tr( ϕ3) theory, Phys. Rev. Lett. 134 (2025) 031601, [ arXiv:2406.04234]

    work page arXiv 2025

  15. [15]

    Bartsch, T

    C. Bartsch, T. V. Brown, K. Kampf, U. Oktem, S. Paranjape, and J. Trnka, Hidden Amplitude Zeros From Double Copy, arXiv:2403.10594

    work page arXiv

  16. [16]

    Y. Li, D. Roest, and T. ter Veldhuis, Hidden Zeros in Scaffolded General Relativity and Exceptional Field Theories, arXiv:2403.12939

    work page arXiv

  17. [17]

    Zhang, New Factorizations of Yang-Mills Amplitudes , arXiv:2406.08969

    Y. Zhang, New Factorizations of Yang-Mills Amplitudes , arXiv:2406.08969

    work page arXiv

  18. [18]

    Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams

    K. Zhou, Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams , arXiv:2411.07944. – 21 –

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    Zhang, On the New Factorizations of Yang-Mills Amplitudes , arXiv:2412.15198

    Y. Zhang, On the New Factorizations of Yang-Mills Amplitudes , arXiv:2412.15198

    work page arXiv

  20. [20]

    Q. Cao, J. Dong, S. He, and C. Shi, A universal splitting of tree-level string and particle scattering amplitudes , Phys. Lett. B 856 (2024) 138934, [ arXiv:2403.08855]

    work page arXiv 2024

  21. [21]

    Arkani-Hamed and C

    N. Arkani-Hamed and C. Figueiredo, All-order splits and multi-soft limits for particle and string amplitudes , arXiv:2405.09608

    work page arXiv

  22. [22]

    Q. Cao, J. Dong, S. He, C. Shi, and F. Zhu, On universal splittings of tree-level particle and string scattering amplitudes, JHEP 09 (2024) 049, [ arXiv:2406.03838]

    work page arXiv 2024

  23. [23]

    New Relations for Einstein-Yang-Mills Amplitudes

    S. Stieberger and T. R. Taylor, New relations for Einstein–Yang–Mills amplitudes , Nucl. Phys. B 913 (2016) 151–162, [arXiv:1606.09616]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  24. [24]

    Amplitude relations in heterotic string theory and Einstein-Yang-Mills

    O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills , JHEP 11 (2016) 074, [arXiv:1608.00130]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  25. [25]

    Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy

    M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy , JHEP 07 (2017) 002, [ arXiv:1703.00421]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Einstein-Yang-Mills from pure Yang-Mills amplitudes

    D. Nandan, J. Plefka, O. Schlotterer, and C. Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes , JHEP 10 (2016) 070, [ arXiv:1607.05701]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  27. [27]

    Relations for Einstein-Yang-Mills amplitudes from the CHY representation

    L. de la Cruz, A. Kniss, and S. Weinzierl, Relations for Einstein–Yang–Mills amplitudes from the CHY representation, Phys. Lett. B 767 (2017) 86–90, [ arXiv:1607.06036]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  28. [28]

    Expansion of Einstein-Yang-Mills Amplitude

    C.-H. Fu, Y.-J. Du, R. Huang, and B. Feng, Expansion of Einstein-Yang-Mills Amplitude , JHEP 09 (2017) 021, [arXiv:1702.08158]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  29. [29]

    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 05 (2017) 075, [arXiv:1703.01269]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  30. [30]

    BCJ numerators from reduced Pfaffian

    Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033, [ arXiv:1703.05717]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  31. [31]

    Y.-J. Du, B. Feng, and F. Teng, Expansion of All Multitrace Tree Level EYM Amplitudes , JHEP 12 (2017) 038, [arXiv:1708.04514]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  32. [32]

    B. Feng, X. Li, and K. Zhou, Expansion of Einstein-Yang-Mills theory by differential operators , Phys. Rev. D 100 (2019), no. 12 125012, [ arXiv:1904.05997]

    work page arXiv 2019

  33. [33]

    Expansion of tree amplitudes for EM and other theories,

    K. Zhou and S.-Q. Hu, Expansions of tree amplitudes for Einstein–Maxwell and other theories , PTEP 2020 (2020), no. 7 073B10, [ arXiv:1907.07857]

    work page arXiv 2020

  34. [34]

    Zhou, Unified web for expansions of amplitudes , JHEP 10 (2019) 195, [ arXiv:1908.10272]

    K. Zhou, Unified web for expansions of amplitudes , JHEP 10 (2019) 195, [ arXiv:1908.10272]

    work page arXiv 2019

  35. [35]

    Expanding single trace YMS amplitudes with gauge invariant coefficients

    F.-S. Wei and K. Zhou, Expanding single-trace YMS amplitudes with gauge-invariant coefficients , Eur. Phys. J. C 84 (2024), no. 1 29, [ arXiv:2306.14774]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  36. [36]

    Recursive construction for expansions of tree Yang-Mills amplitudes from soft theorem

    C. Hu and K. Zhou, Recursive construction for expansions of tree Yang–Mills amplitudes from soft theorem , Eur. Phys. J. C 84 (2024), no. 3 221, [ arXiv:2311.03112]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  37. [37]

    Multi-trace YMS amplitudes from soft behavior

    Y.-J. Du and K. Zhou, Multi-trace YMS amplitudes from soft behavior , JHEP 03 (2024) 081, [arXiv:2401.03879]. – 22 –

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  38. [38]

    Towards tree Yang-Mills and Yang-Mills-scalar amplitudes with higher-derivative interactions

    K. Zhou and C. Hu, Towards tree Yang-Mills and Yang-Mills-scalar amplitudes with higher-derivative interactions, JHEP 01 (2025) 167, [ arXiv:2406.03034]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  39. [39]

    Constructing tree amplitudes of scalar EFT from double soft theorem

    K. Zhou, Constructing tree amplitudes of scalar EFT from double soft theorem , JHEP 12 (2024) 079, [arXiv:2406.03784]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  40. [40]

    The galileon as a local modification of gravity

    A. Nicolis, R. Rattazzi, and E. Trincherini, The Galileon as a local modification of gravity , Phys. Rev. D 79 (2009) 064036, [ arXiv:0811.2197]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  41. [41]

    A. A. Tseytlin, Born-Infeld action, supersymmetry and string theory , hep-th/9908105

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [42]

    Kleiss and H

    R. Kleiss and H. Kuijf, Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders , Nucl. Phys. B 312 (1989) 616–644

    work page 1989

  43. [43]

    F. A. Berends and W. T. Giele, Recursive Calculations for Processes with n Gluons , Nucl. Phys. B 306 (1988) 759–808. – 23 –

    work page 1988