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arxiv: 2306.14774 · v2 · submitted 2023-06-26 · ✦ hep-th

Expanding single trace YMS amplitudes with gauge invariant coefficients

Fang-Stars Wei , Kang Zhou This is my paper

Pith reviewed 2026-05-24 08:21 UTC · model grok-4.3

classification ✦ hep-th
keywords single-trace Yang-Mills-scalar amplitudessoft theoremsgauge invariancepermutation symmetrycolor-kinematic dualityrecursive expansionbottom-up construction
0
0 comments X

The pith

A recursive construction of single-trace Yang-Mills-scalar amplitudes from soft theorems produces coefficients that are gauge invariant for any gluon polarization and symmetric under gluon permutations.

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

The paper builds the expansion of single-trace Yang-Mills-scalar amplitudes step by step using only soft theorems. Each added term is fixed so that the full expression remains consistent with gauge transformations on the gluons and with exchanging any two gluons. Because the construction never introduces non-invariant pieces, the final result automatically satisfies the same properties that were previously obtained through covariant color-kinematic duality. A reader cares because the method supplies an explicit, symmetry-manifest expansion that can be written down for arbitrary numbers of external legs without first solving the full scattering equations or invoking duality maps.

Core claim

The expansion of single-trace Yang-Mills-scalar amplitudes constructed recursively using soft theorems manifests the gauge invariance for any polarization carried by external gluons, as well as the permutation symmetry among external gluons, and is equivalent to the result found via covariant color-kinematic duality.

What carries the argument

Recursive soft-theorem construction that determines each higher-point term from the lower-point amplitude while preserving gauge invariance and gluon exchange symmetry at every step.

If this is right

  • Every coefficient in the resulting expansion is gauge invariant under shifts of any gluon polarization vector.
  • The expression is manifestly symmetric under arbitrary permutations of the external gluons.
  • The same recursive procedure reproduces the known result obtained from covariant color-kinematic duality.
  • The construction extends directly to higher-point single-trace amplitudes without additional symmetry constraints.

Where Pith is reading between the lines

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

  • The same soft-theorem recursion could be applied to multi-trace or other extended Yang-Mills-scalar theories where color-kinematic duality is harder to implement.
  • Because the method never needs to impose gauge invariance by hand, it may reduce the algebraic complexity of writing down explicit amplitudes for large numbers of legs.
  • Testing the recursion on a five-point single-trace example would show whether the pattern of coefficients continues to match duality-based expressions term by term.

Load-bearing premise

Soft theorems alone are enough to fix every coefficient in the expansion while automatically keeping the whole expression gauge invariant and permutation symmetric.

What would settle it

Compute the four-gluon two-scalar single-trace amplitude both by the recursive soft-theorem method and by the covariant color-kinematic duality method; the two expressions must be identical after simplification.

Figures

Figures reproduced from arXiv: 2306.14774 by Fang-Stars Wei, Kang Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1: Two 5-point diagrams [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Turn the 4-point vertex to 3-point ones. The bold line corresponds to the inserted propa [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

In this note, we use the new bottom up method based on soft theorems to construct the expansion of single-trace Yang-Mills-scalar amplitudes recursively. The resulted expansion manifests the gauge invariance for any polarization carried by external gluons, as well as the permutation symmetry among external gluons. Our result is equivalent to that found by Clifford Cheung and James Mangan via the so called covariant color-kinematic duality approach.

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 presents a bottom-up recursive construction of single-trace Yang-Mills-scalar (YMS) amplitudes using soft theorems. It asserts that the resulting expansion automatically manifests gauge invariance under arbitrary gluon polarizations and permutation symmetry among external gluons, and that the expansion is equivalent to the one previously obtained by Cheung and Mangan via covariant color-kinematic duality.

Significance. If the recursive procedure is shown to preserve the claimed symmetries at every step without additional assumptions or contact-term ambiguities, the work supplies an independent derivation of the YMS expansion that directly ties the result to soft theorems. This would strengthen the understanding of how soft limits enforce gauge invariance and could serve as a template for similar constructions in other theories with mixed particle content. The claimed equivalence to the covariant CK-duality result is a useful cross-check, though its value depends on the explicit verification provided in the manuscript.

major comments (2)
  1. The central claim that the recursive construction 'manifests' gauge invariance for any gluon polarization is stated in the abstract but is not supported by an explicit inductive argument. The manuscript must demonstrate that multiplication by the soft factor preserves the Ward identity (vanishing under ε_i → k_i) when the lower-point amplitude satisfies it, including a check that no regular (non-soft) contact terms are introduced that could violate the identity for generic polarizations. Without this step, the assertion remains unproven.
  2. The abstract asserts that the construction yields permutation symmetry among external gluons, yet no base cases (e.g., 3- or 4-point amplitudes) or explicit recursion relation are supplied to verify that the symmetry is inherited at each step. Soft theorems determine amplitudes only up to regular terms; an explicit demonstration that the chosen normalization and recursion fix these terms while preserving the symmetry is required for the claim to be load-bearing.
minor comments (2)
  1. The abstract refers to 'the new bottom up method based on soft theorems' without citing the specific prior reference that defines the method or the precise form of the soft factors employed.
  2. The equivalence to the Cheung-Mangan result is asserted but not accompanied by a side-by-side comparison of coefficients or an explicit statement of the normalization convention used in both approaches.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the recommendation for major revision. The comments correctly identify that the manuscript would benefit from explicit inductive arguments and base-case verifications to support the claims of gauge invariance and permutation symmetry. We will revise the paper accordingly.

read point-by-point responses

  1. Referee: The central claim that the recursive construction 'manifests' gauge invariance for any gluon polarization is stated in the abstract but is not supported by an explicit inductive argument. The manuscript must demonstrate that multiplication by the soft factor preserves the Ward identity (vanishing under ε_i → k_i) when the lower-point amplitude satisfies it, including a check that no regular (non-soft) contact terms are introduced that could violate the identity for generic polarizations. Without this step, the assertion remains unproven.

    Authors: We agree that an explicit inductive proof is needed for rigor. In the revised version we will add a dedicated subsection proving by induction that if the lower-point single-trace YMS amplitude satisfies the Ward identity for all gluon polarizations, then the soft-factor multiplication (with the contact terms fixed by the recursion) also satisfies it for the new gluon. The argument will explicitly verify that the soft factor itself vanishes under ε → k and that no additional regular terms violating the identity are generated for generic polarizations. Base cases (3- and 4-point) will be checked directly. revision: yes

  2. Referee: The abstract asserts that the construction yields permutation symmetry among external gluons, yet no base cases (e.g., 3- or 4-point amplitudes) or explicit recursion relation are supplied to verify that the symmetry is inherited at each step. Soft theorems determine amplitudes only up to regular terms; an explicit demonstration that the chosen normalization and recursion fix these terms while preserving the symmetry is required for the claim to be load-bearing.

    Authors: We accept that the current text does not supply the required explicit checks. The revised manuscript will include the explicit recursion relation, the 3- and 4-point base cases (computed both from the soft theorem and by direct Feynman rules), and an inductive step showing that the chosen normalization of the regular terms preserves full permutation symmetry among the gluons. This will also clarify how the soft-theorem ambiguities are fixed without breaking the symmetry. revision: yes

Circularity Check

0 steps flagged

No circularity: recursive construction from external soft theorems matches independent reference

full rationale

The paper constructs single-trace YMS amplitudes recursively from soft theorems and asserts that the resulting expansion manifests gauge invariance and permutation symmetry while being equivalent to the independent result of Cheung and Mangan via covariant color-kinematic duality. No quoted equation or step reduces the claimed properties to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation is presented as self-contained against the external benchmark, consistent with a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the construction relies on standard soft theorems whose status is not detailed here.

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discussion (0)

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Forward citations

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