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arxiv: 2212.12892 · v2 · submitted 2022-12-25 · ✦ hep-th

Tree level amplitudes from soft theorems

Kang Zhou This is my paper

Pith reviewed 2026-05-24 10:16 UTC · model grok-4.3

classification ✦ hep-th
keywords tree-level amplitudessoft theoremsYang-Millsdouble copysoft factorsEinstein-Yang-Millsgraviton
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0 comments X

The pith

Soft theorems plus universality fix the explicit form of tree-level amplitudes in Yang-Mills and gravity theories.

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

The paper establishes that the behavior of amplitudes under soft limits, when combined with the requirement that soft factors take the same universal form in every amplitude, determines both the amplitudes themselves and the explicit expressions for those soft factors. Imposing these conditions together with the double-copy relation allows reconstruction of the expanded single-trace Yang-Mills-scalar and pure Yang-Mills amplitudes. The same conditions fix the soft factors for the bi-adjoint scalar and the gluon, and the construction extends directly to Einstein-Yang-Mills and gravitational amplitudes, reproducing the known graviton soft factors. A reader would care because the result offers a route to amplitudes that begins from their infrared behavior rather than from a Lagrangian or Feynman rules.

Core claim

By imposing the soft theorems and the universality of soft factors, together with the assumption of double copy, the single-trace Yang-Mills-scalar amplitudes and pure Yang-Mills amplitudes are reconstructed in expanded form; the explicit soft factors for the bi-adjoint scalar and the gluon are thereby determined; the expansions extend to Einstein-Yang-Mills and gravitational amplitudes, from which the graviton soft factors are recovered.

What carries the argument

Soft theorems combined with universality of soft factors, used under the double-copy assumption to reconstruct amplitudes.

If this is right

  • Tree-level single-trace Yang-Mills-scalar and pure Yang-Mills amplitudes are fixed once soft theorems and universality are imposed.
  • Explicit formulas for the bi-adjoint scalar and gluon soft factors are obtained directly from the same conditions.
  • The same procedure reproduces the graviton soft factors from the expanded Einstein-Yang-Mills amplitudes.
  • The expansions of Yang-Mills-scalar and Yang-Mills amplitudes extend consistently to Einstein-Yang-Mills and gravitational amplitudes.

Where Pith is reading between the lines

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

  • The method supplies a possible alternative starting point for deriving amplitudes in theories whose soft theorems are known but whose Lagrangians are not.
  • If the uniqueness result holds, the double-copy relation itself may be viewed as a consequence of soft universality rather than an independent input.
  • The construction could be tested by checking whether the reconstructed amplitudes match known results at higher multiplicity or in other kinematic limits.

Load-bearing premise

Soft factors must take exactly the same explicit form in every amplitude of a given theory.

What would settle it

An explicit amplitude (or set of amplitudes) that satisfies the same soft theorems and universality condition yet differs from the reconstructed expressions would falsify the uniqueness claim.

read the original abstract

We demonstrate that the tree level amplitudes and the explicit formulas of soft factors can be uniquely determined by soft theorems and the universality of soft factors. By imposing the soft theorems and the universality, as well as the assumption of double copy, we reconstruct single trace Yang-Millsscalar amplitudes and pure Yang-Mills amplitudes, in the expanded formulas. The explicit formulas of soft factors for the bi-adjoint scalar and gluon are also determined. The expansions of Yang-Mills-scalar and Yang-Mills amplitudes can be extended to Einstein-Yang-Mills and gravitational amplitudes, and we use the expanded single trace Einstein-Yang-Mills amplitudes to reproduce the soft factors for the graviton.

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 that tree-level amplitudes and explicit soft factor formulas are uniquely determined by soft theorems together with universality of soft factors. Imposing these conditions plus the double-copy assumption, the authors reconstruct expanded single-trace Yang-Mills-scalar and pure Yang-Mills amplitudes, obtain explicit soft factors for the bi-adjoint scalar and gluon, and extend the construction to Einstein-Yang-Mills and gravitational amplitudes, using the former to recover the graviton soft factor.

Significance. If amplitudes could be uniquely fixed from soft theorems and universality alone, the result would supply a new constructive principle in the amplitudes literature. The manuscript does deliver explicit expanded formulas once the double-copy assumption is added, but this external input is required to close the system.

major comments (2)
  1. [Abstract] Abstract: the statement that amplitudes 'can be uniquely determined by soft theorems and the universality of soft factors' is not supported by the subsequent reconstruction, which requires the additional double-copy assumption to obtain explicit formulas.
  2. [Reconstruction section] Reconstruction of single-trace Yang-Mills-scalar and pure Yang-Mills amplitudes: these are obtained only after the double-copy assumption is imposed; the soft theorems and universality conditions alone leave the amplitudes under-determined, so the uniqueness claim rests on an external input not derived from the stated principles.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the assumptions underlying our uniqueness claims. We agree that revisions are required to make the role of the double-copy assumption explicit in both the abstract and the reconstruction section.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that amplitudes 'can be uniquely determined by soft theorems and the universality of soft factors' is not supported by the subsequent reconstruction, which requires the additional double-copy assumption to obtain explicit formulas.

    Authors: We acknowledge that the abstract's opening claim is imprecise as written. The uniqueness statement applies when the double-copy assumption is included alongside the soft theorems and universality conditions. We will revise the abstract to state explicitly: 'We demonstrate that the tree level amplitudes and the explicit formulas of soft factors can be uniquely determined by soft theorems, the universality of soft factors, and the double-copy assumption.' revision: yes

  2. Referee: [Reconstruction section] Reconstruction of single-trace Yang-Mills-scalar and pure Yang-Mills amplitudes: these are obtained only after the double-copy assumption is imposed; the soft theorems and universality conditions alone leave the amplitudes under-determined, so the uniqueness claim rests on an external input not derived from the stated principles.

    Authors: The referee correctly notes that soft theorems and universality alone leave the amplitudes under-determined. The double-copy assumption supplies the additional relations required to close the system and obtain explicit formulas. In the revised manuscript we will add a clarifying paragraph in the reconstruction section stating that the soft theorems and universality conditions are insufficient by themselves and that the double-copy input is essential for uniqueness. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation incorporates external double-copy assumption without self-referential reduction

full rationale

The paper states that amplitudes and soft factors are reconstructed by imposing soft theorems, universality of soft factors, and the double-copy assumption. This external input (double copy) supplies the additional structure needed for explicit formulas, but the steps do not reduce any claimed prediction to a tautology or to a fitted parameter renamed as output. No self-definitional equations, no fitted-input-called-prediction pattern, and no load-bearing self-citations are exhibited in the provided text. The derivation chain therefore remains self-contained relative to its stated inputs and does not meet the criteria for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper's central claim depends on these three assumptions from the field of scattering amplitudes; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption Soft theorems apply to the tree-level amplitudes considered
    Imposed as the starting point to determine the amplitudes uniquely
  • domain assumption Soft factors are universal across different amplitudes
    Used together with soft theorems to fix the explicit formulas
  • domain assumption Double copy relation holds between the relevant gauge and gravity theories
    Assumed to extend the reconstruction from Yang-Mills to Einstein-Yang-Mills and gravity

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discussion (0)

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Forward citations

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