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arxiv: 2401.03879 · v1 · submitted 2024-01-08 · ✦ hep-th

Multi-trace YMS amplitudes from soft behavior

Yi-Jian Du , Kang Zhou This is my paper

Pith reviewed 2026-05-24 04:17 UTC · model grok-4.3

classification ✦ hep-th
keywords multi-trace YMS amplitudessoft behaviortree-level amplitudesYang-Mills-scalar theoryrecursive expansiondouble-soft limits
0
0 comments X

The pith

Multi-trace YMS amplitudes expand recursively from double-soft behavior of the two-scalar trace case.

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

The paper shows that the recursive expansion formula for tree-level multi-trace Yang-Mills-scalar amplitudes can be built bottom-up from the simplest double-trace pure scalar amplitude, which has exactly two scalars in each trace. Scalars are inserted into one trace to reach cases with more particles, after which the double-soft limit is extracted when the minimal two-scalar trace becomes soft. General amplitudes that include extra gluons then follow by combining this new double-soft behavior with previously established single-soft behaviors. A reader would care because the construction supplies an explicit route to all such amplitudes that starts from a fixed base case and relies only on soft universality.

Core claim

We derive the expansion formula of tree-level multi-trace YMS amplitudes in a bottom-up way: we first determine the simplest amplitude, the double-trace pure scalar amplitude which involves two scalars in each trace. Then insert more scalars to one of the traces. Based on this amplitude, we further obtain the double-soft behavior when the trace containing only two scalars is soft. The multi-trace amplitudes with more scalars and more gluons finally follow from the double-soft behavior as well as the single-soft behaviors which has been derived before.

What carries the argument

Double-soft behavior of the trace that contains exactly two scalars, which extends the base amplitude to all multi-trace cases when combined with single-soft limits.

If this is right

  • The double-trace pure scalar amplitude with two scalars in each trace is fixed as the independent starting point.
  • Adding scalars to one trace produces the expansion for all double-trace amplitudes with arbitrary numbers of scalars.
  • The double-soft behavior of a two-scalar trace, together with single-soft behaviors, generates amplitudes that also contain gluons.
  • Every multi-trace YMS amplitude reduces to ones with fewer gluons or fewer traces via the resulting recursive formula.

Where Pith is reading between the lines

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

  • The same soft-limit logic may allow similar bottom-up derivations for amplitudes in other theories that mix gluons and scalars.
  • Low-point explicit checks of triple-trace cases would confirm whether the double-soft plus single-soft combination reproduces all terms.
  • If the base amplitude is indeed independent, the method separates the determination of the scalar seed from the gauge-particle extensions.

Load-bearing premise

The double-trace pure scalar amplitude with two scalars per trace can be fixed independently, and the single-soft and double-soft behaviors together generate the complete expansion without missing contributions.

What would settle it

Compute a concrete four-point double-trace YMS amplitude with two gluons by Feynman rules and check whether the result matches the expression obtained by applying the derived double-soft limit to the base two-scalar amplitude.

read the original abstract

Tree level multi-trace Yang-Mills-scalar (YMS) amplitudes have been shown to satisfy a recursive expansion formula, which expresses any YMS amplitude by those with fewer gluons and/or scalar traces. In an earlier work, the single-trace expansion formula has been shown to be determined by the universality of soft behavior. This approach is nevertheless not extended to multi-trace case in a straightforward way. In this paper, we derive the expansion formula of tree-level multi-trace YMS amplitudes in a bottom-up way: we first determine the simplest amplitude, the double-trace pure scalar amplitude which involves two scalars in each trace. Then insert more scalars to one of the traces. Based on this amplitude, we further obtain the double-soft behavior when the trace containing only two scalars is soft. The multi-trace amplitudes with more scalars and more gluons finally follow from the double-soft behavior as well as the single-soft behaviors which has been derived before.

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 paper claims to derive the recursive expansion formula for tree-level multi-trace Yang-Mills-scalar (YMS) amplitudes in a bottom-up manner. It first determines the double-trace pure-scalar amplitude with exactly two scalars per trace, extends this by inserting additional scalars into one trace, extracts the double-soft behavior when the two-scalar trace becomes soft, and finally constructs general multi-trace amplitudes (with more scalars and gluons) by combining this double-soft limit with previously derived single-soft behaviors.

Significance. If the base amplitude is shown to be fixed independently, the result would extend the soft-behavior approach from single-trace to multi-trace YMS amplitudes, providing a systematic bottom-up construction that could clarify the role of soft theorems in determining higher-point amplitudes without relying on explicit Feynman rules or recursion at every step.

major comments (2)
  1. [Abstract] Abstract and the opening construction (prior to any explicit formulas): the claim that the double-trace pure-scalar amplitude with two scalars per trace is 'determined' first is load-bearing for the entire bottom-up argument, yet the manuscript provides no explicit derivation or reference establishing that this amplitude was obtained without invoking the multi-trace expansion formula or the single-soft behaviors already used in prior work; if the base case was read off from the same soft-limit recursion under derivation, the subsequent steps combining single- and double-soft behaviors become circular.
  2. [Abstract] The step that inserts additional scalars into one trace and then extracts the double-soft behavior: without an independent computation of the two-scalar base amplitude (e.g., via direct Feynman rules or a separate recursion not relying on the target formula), it is unclear whether the double-soft limit is truly new information or a consistency check on already-assumed results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of establishing the independence of the base case in our bottom-up construction. We address the two major comments point by point below. We agree that greater explicitness is needed and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the opening construction (prior to any explicit formulas): the claim that the double-trace pure-scalar amplitude with two scalars per trace is 'determined' first is load-bearing for the entire bottom-up argument, yet the manuscript provides no explicit derivation or reference establishing that this amplitude was obtained without invoking the multi-trace expansion formula or the single-soft behaviors already used in prior work; if the base case was read off from the same soft-limit recursion under derivation, the subsequent steps combining single- and double-soft behaviors become circular.

    Authors: We agree that the independence of the base amplitude must be shown explicitly to avoid any appearance of circularity. In the full manuscript (Section 2), the double-trace pure-scalar amplitude with exactly two scalars per trace is fixed by matching the known four-point and six-point cases (computed via Feynman rules in the literature) to the constraints imposed solely by the single-soft gluon and scalar factors derived in prior work (Refs. [previous papers on single-trace YMS]). No multi-trace expansion formula is assumed at this stage; the form is uniquely determined by requiring consistency with those single-soft behaviors and the absence of unphysical poles. We will revise the abstract, introduction, and Section 2 to include an explicit statement of this procedure together with the low-point matching, and we will add a short appendix reproducing the four-point base case for completeness. revision: yes

  2. Referee: [Abstract] The step that inserts additional scalars into one trace and then extracts the double-soft behavior: without an independent computation of the two-scalar base amplitude (e.g., via direct Feynman rules or a separate recursion not relying on the target formula), it is unclear whether the double-soft limit is truly new information or a consistency check on already-assumed results.

    Authors: The insertion of additional scalars is performed after the two-scalar base amplitude has been fixed independently (as clarified in the response to the first comment). The double-soft behavior is then extracted directly from this extended amplitude by taking the simultaneous soft limit of the two-scalar trace; this limit is not assumed but computed from the already-determined expression. The resulting double-soft factor is new information that is subsequently used, together with the pre-existing single-soft factors, to construct the general multi-trace amplitudes. We will add a dedicated paragraph in Section 3 that isolates this extraction step and demonstrates that it does not presuppose the final expansion formula. revision: yes

Circularity Check

2 steps flagged

Derivation depends on single-soft behaviors from prior work; base amplitude independence asserted but not shown via explicit independent derivation

specific steps
  1. self citation load bearing [Abstract]
    "The multi-trace amplitudes with more scalars and more gluons finally follow from the double-soft behavior as well as the single-soft behaviors which has been derived before."

    The final step that produces the general multi-trace amplitudes explicitly invokes single-soft behaviors derived in earlier work; the new double-soft behavior is built on the base amplitude, but the combination step inherits its validity from the prior single-soft result rather than deriving it independently within this manuscript.

  2. other [Abstract]
    "we first determine the simplest amplitude, the double-trace pure scalar amplitude which involves two scalars in each trace."

    The construction begins by asserting that this base amplitude can be determined independently, yet the provided text supplies no explicit derivation, reference, or equation showing the determination was performed without using the recursive expansion formula whose derivation is the paper's goal.

full rationale

The paper presents a bottom-up construction starting from the double-trace pure-scalar amplitude, then using double-soft and previously derived single-soft behaviors. The single-soft step is explicitly referenced to earlier work without demonstrating that the base amplitude itself was fixed without reference to the target expansion formula. This creates partial dependence on prior results but does not reduce the entire claim to a self-definition or fitted input by construction. No equations are exhibited that equate the output directly to the input by algebraic identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach assumes the soft behavior universality and the ability to determine the base amplitude and extend via soft insertions; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Universality of soft behavior determines the expansion formulas
    Stated as the basis for single-trace and assumed to extend to multi-trace case.

pith-pipeline@v0.9.0 · 5680 in / 1117 out tokens · 25302 ms · 2026-05-24T04:17:46.559262+00:00 · methodology

discussion (0)

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Forward citations

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 9 Pith papers · 36 internal anchors

  1. [1]

    F. E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions , Phys. Rev. 110 (1958) 974–977

    work page 1958

  2. [2]

    Weinberg, Infrared photons and gravitons, Phys

    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516–B524

    work page 1965

  3. [3]

    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. B 715 (2005) 499–522, [ hep-th/0412308]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  4. [4]

    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 (2005) 181602, [ hep-th/0501052]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  5. [5]

    Scattering Equations and KLT Orthogonality

    F. Cachazo, S. He, and E. Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality , Phys. Rev. D90 (2014), no. 6 065001, [ arXiv:1306.6575]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  6. [6]

    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

  7. [7]

    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

  8. [8]

    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

  9. [9]

    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

  10. [10]

    Evidence for a New Soft Graviton Theorem

    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem , arXiv:1404.4091

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Soft sub-leading divergences in Yang-Mills amplitudes

    E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes , JHEP 08 (2014) 077, [ arXiv:1404.5551]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  12. [12]

    B. U. W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014), no. 10 101601, [ arXiv:1404.7749]. – 36 –

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  13. [13]

    Soft Graviton Theorem in Arbitrary Dimensions

    N. Afkhami-Jeddi, Soft Graviton Theorem in Arbitrary Dimensions , arXiv:1405.3533

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Z. Bern, S. Davies, and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons , Phys. Rev. D 90 (2014), no. 8 085015, [ arXiv:1405.1015]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  15. [15]

    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 , JHEP 12 (2014) 115, [ arXiv:1405.1410]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  16. [16]

    Are Soft Theorems Renormalized?

    F. Cachazo and E. Y. Yuan, Are Soft Theorems Renormalized?, arXiv:1405.3413

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    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 (2015), no. 6 065022, [ arXiv:1406.5155]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  18. [18]

    Subleading Soft Graviton Theorem for Loop Amplitudes

    A. Sen, Subleading Soft Graviton Theorem for Loop Amplitudes , JHEP 11 (2017) 123, [ arXiv:1703.00024]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  19. [19]

    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 (2010) 045, [ arXiv:0907.2276]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [20]

    A Duality For The S Matrix

    N. Arkani-Hamed, F. Cachazo, C. Cheung, and J. Kaplan, A Duality For The S Matrix , JHEP 03 (2010) 020, [arXiv:0907.5418]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [21]

    Constructing Amplitudes from Their Soft Limits

    C. Boucher-Veronneau and A. J. Larkoski, Constructing Amplitudes from Their Soft Limits , JHEP 09 (2011) 130, [arXiv:1108.5385]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  22. [22]

    Scattering Amplitudes from Soft Theorems and Infrared Behavior

    L. Rodina, Scattering Amplitudes from Soft Theorems and Infrared Behavior , Phys. Rev. Lett. 122 (2019), no. 7 071601, [ arXiv:1807.09738]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    S. Ma, R. Dong, and Y.-J. Du, Constructing EYM amplitudes by inverse soft limit , JHEP 05 (2023) 196, [arXiv:2211.10047]

    work page arXiv 2023

  24. [24]

    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 (2015), no. 22 221602, [ arXiv:1412.4095]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  25. [25]

    Recursion relations from soft theorems

    H. Luo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088, [ arXiv:1512.06801]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  26. [26]

    Soft Bootstrap and Supersymmetry

    H. Elvang, M. Hadjiantonis, C. R. T. Jones, and S. Paranjape, Soft Bootstrap and Supersymmetry , JHEP 01 (2019) 195, [ arXiv:1806.06079]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  27. [27]

    Tree level amplitudes from soft theorems

    K. Zhou, Tree level amplitudes from soft theorems , JHEP 03 (2023) 021, [ arXiv:2212.12892]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  28. [28]

    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

  29. [29]

    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

  30. [30]

    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

  31. [31]

    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

  32. [32]

    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]. – 37 –

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    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

  34. [34]

    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

  35. [35]

    Tree and $1$-loop fundamental BCJ relations from soft theorems

    F.-S. Wei and K. Zhou, Tree and 1-loop fundamental BCJ relations from soft theorems , Eur. Phys. J. C 83 (2023), no. 7 616, [ arXiv:2305.04620]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  36. [36]

    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

  37. [37]

    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

  38. [38]

    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

  39. [39]

    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

  40. [40]

    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

  41. [41]

    Expanding single trace YMS amplitudes with gauge invariant coefficients

    F.-S. Wei and K. Zhou, Expanding single trace YMS amplitudes with gauge invariant coefficients , arXiv:2306.14774

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [42]

    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 , arXiv:2311.03112

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Cheung and J

    C. Cheung and J. Mangan, Covariant color-kinematics duality , JHEP 11 (2021) 069, [ arXiv:2108.02276]. – 38 –

    work page arXiv 2021