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arxiv: 2506.00747 · v1 · submitted 2025-05-31 · ✦ hep-th

Soft theorems of tree-level {rm Tr}(φ³), YM and NLSM amplitudes from 2-splits

Kang Zhou This is my paper

Pith reviewed 2026-05-19 11:40 UTC · model grok-4.3

classification ✦ hep-th
keywords soft theoremstree-level amplitudesTr(phi^3)Yang-Millsnonlinear sigma modelfactorization2-splitskinematic space
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0 comments X

The pith

Factorization and 2-splits reproduce soft theorems for tree-level Tr(φ³), YM and NLSM amplitudes.

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

The paper establishes that soft theorems arise directly from factorization properties of scattering amplitudes. It combines standard factorizations at physical poles with newly identified 2-splits that occur at special points in the space of momenta. This reproduces the leading and next-to-leading single-soft theorems for Tr(φ³) and Yang-Mills amplitudes at tree level. The same method gives the leading and next-to-leading double-soft theorems for the nonlinear sigma model. The soft factors remain consistent when momenta are rearranged using conservation laws.

Core claim

Using only factorization properties including conventional factorizations on physical poles as well as 2-splits on special loci in kinematic space, the leading and sub-leading single-soft theorems for tree-level Tr(φ³) and Yang-Mills amplitudes are reproduced, together with the leading and sub-leading double-soft theorems for tree-level NLSM amplitudes.

What carries the argument

2-splits, which factorize an amplitude into a product of two lower-point amplitudes on special loci in kinematic space.

Load-bearing premise

The 2-split factorizations must hold on the specified special loci in kinematic space.

What would settle it

Explicitly compute the soft expansion of a five-point tree-level Yang-Mills amplitude and check whether the sub-leading term matches the derived soft factor from the 2-split method.

read the original abstract

In this paper, we extend the method proposed in \cite{Arkani-Hamed:2024fyd} for deriving soft theorems of amplitudes, which relies exclusively on factorization properties including conventional factorizations on physical poles, as well as newly discovered $2$-splits on special loci in kinematic space. Using the extended approach, we fully reproduce the leading and sub-leading single-soft theorems for tree-level ${\rm Tr}(\phi^3)$ and Yang-Mills (YM) amplitudes, along with the leading and sub-leading double-soft theorems for tree-level amplitudes of non-linear sigma model (NLSM). Furthermore, we establish universal representations of higher-order single-soft theorems for tree-level ${\rm Tr}(\phi^3)$ and YM amplitudes in reduced lower-dimensional kinematic spaces. All obtained soft factors maintain consistency with momentum conservation; that is, while each explicit expression of the resulting soft behavior may changes under re-parameterization via momentum conservation, the physical content remains equivalent. Additionally, we find two interesting by-products: First, the single-soft theorems of YM amplitudes and the double-soft theorems of NLSM, at leading and sub-leading orders, are related by a simple kinematic replacement. This replacement also transmutes gauge invariance to Adler zero. Second, we obtain universal sub-leading soft theorems for the resulting pure YM and NLSM currents in the corresponding $2$-splits.

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

1 major / 4 minor

Summary. The manuscript extends the 2-split factorization approach introduced in Arkani-Hamed:2024fyd, combining conventional factorizations on physical poles with 2-splits on special kinematic loci to derive soft theorems. It reproduces the leading and sub-leading single-soft theorems for tree-level Tr(φ³) and Yang-Mills amplitudes, as well as the leading and sub-leading double-soft theorems for tree-level NLSM amplitudes. Additional results include universal representations of higher-order single-soft theorems in reduced lower-dimensional kinematic spaces, verification that all soft factors are consistent with momentum conservation under re-parameterization, a kinematic replacement relating YM single-soft and NLSM double-soft theorems (transmuting gauge invariance to the Adler zero), and universal sub-leading soft theorems for the pure YM and NLSM currents arising in the 2-splits.

Significance. If the 2-split relations hold on the stated loci, the work supplies a purely factorization-based route to soft theorems that unifies Tr(φ³), YM, and NLSM without invoking Feynman rules, BCFW recursion, or other standard techniques. The explicit kinematic replacement connecting single-soft YM to double-soft NLSM, together with the resulting transmutation of gauge invariance into the Adler zero, is a concrete and falsifiable by-product. The consistency checks under momentum conservation and the derivation of soft theorems for the associated currents further strengthen the factorization perspective. These elements constitute a non-trivial consistency test of the 2-split method on established results.

major comments (1)
  1. [§2] §2 (Method and 2-split factorizations): The derivations of the soft factors at leading and sub-leading orders rely on the explicit form and validity of the 2-split identities on special kinematic loci, which are imported from Arkani-Hamed:2024fyd without re-derivation or independent verification inside this manuscript. Because these loci are load-bearing for fixing the soft behavior when combined with ordinary pole factorizations, the central reproduction claim is conditional on the correctness and applicability of the external construction; a concise recap of the relevant loci (or the precise equations from the cited reference) for each of the three theories would make the argument self-contained.
minor comments (4)
  1. [Abstract] Abstract: The phrase 'newly discovered 2-splits' could be adjusted to 'recently proposed 2-splits' to reflect that the factorization relations originate in the cited reference rather than being discovered in the present work.
  2. [§4] §4 (Kinematic replacement): The relation between YM single-soft and NLSM double-soft theorems via a simple replacement is highlighted as a by-product; an explicit side-by-side comparison of the soft factors before and after the replacement would clarify how gauge invariance maps to the Adler zero.
  3. [§5] §5 (Universal representations): The reduced lower-dimensional kinematic spaces used for the higher-order single-soft theorems are introduced without a diagram or explicit coordinate choice; a short kinematic sketch would aid readability.
  4. Throughout: Several instances of 'the resulting soft behavior' appear; consistent use of 'soft factor' or 'soft theorem' would improve terminological uniformity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. We address the single major comment below and will revise the manuscript accordingly to improve its self-containment.

read point-by-point responses

  1. Referee: [§2] §2 (Method and 2-split factorizations): The derivations of the soft factors at leading and sub-leading orders rely on the explicit form and validity of the 2-split identities on special kinematic loci, which are imported from Arkani-Hamed:2024fyd without re-derivation or independent verification inside this manuscript. Because these loci are load-bearing for fixing the soft behavior when combined with ordinary pole factorizations, the central reproduction claim is conditional on the correctness and applicability of the external construction; a concise recap of the relevant loci (or the precise equations from the cited reference) for each of the three theories would make the argument self-contained.

    Authors: We agree that a concise recap of the relevant 2-split identities and kinematic loci would enhance the self-contained nature of the argument. In the revised manuscript we will insert a short summary paragraph in Section 2 that states the precise kinematic loci and the corresponding 2-split equations (with equation numbers) from Arkani-Hamed:2024fyd for the Tr(φ³), YM, and NLSM cases. This addition will provide the necessary context for the subsequent soft-theorem derivations while keeping the focus on the new results of the present work. revision: yes

Circularity Check

0 steps flagged

No significant circularity: soft theorems reproduced from external 2-split factorizations

full rationale

The paper extends the factorization-based method of the distinct-authored reference Arkani-Hamed:2024fyd, treating the existence and applicability of 2-splits on special kinematic loci as an external input rather than deriving those loci or the splits from the soft theorems under consideration. Conventional pole factorizations are combined with these imported 2-splits to obtain explicit leading and sub-leading soft factors for Tr(φ³), YM, and NLSM amplitudes, with all expressions verified to be consistent under momentum conservation reparameterizations. Because the 2-splits originate outside the present work and are not shown to be constructed from the target soft behavior, no equation or derivation step reduces the reproduced soft theorems to quantities defined solely by the paper's own inputs or self-referential fits. The central reproduction therefore remains a non-circular application of independent factorization properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations assume standard tree-level factorization on physical poles and the existence of 2-split loci on special kinematic surfaces; no new free parameters are introduced, no new particles or forces are postulated, and the only background assumptions are momentum conservation and the factorization properties imported from the cited reference.

axioms (2)
  • domain assumption Tree-level amplitudes factorize on physical poles and admit 2-split decompositions on special loci in kinematic space.
    Invoked throughout the abstract as the sole basis for deriving the soft theorems; taken from the method of Arkani-Hamed:2024fyd.
  • standard math Momentum conservation holds and can be used to re-parameterize expressions without changing physical content.
    Explicitly stated as a consistency check for all obtained soft factors.

pith-pipeline@v0.9.0 · 5777 in / 1651 out tokens · 38463 ms · 2026-05-19T11:40:51.447846+00:00 · methodology

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Forward citations

Cited by 3 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. 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.

Reference graph

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