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arxiv: 2411.07944 · v1 · submitted 2024-11-12 · ✦ hep-th

Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams

Kang Zhou This is my paper

Pith reviewed 2026-05-23 17:46 UTC · model grok-4.3

classification ✦ hep-th
keywords Feynman diagramshidden zerosamplitude splittingsordered tree amplitudesYang-Millsnon-linear sigma modelTr(phi^3)
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The pith

Feynman diagram cuts in three universal ways explain hidden zeros and splittings of ordered tree 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 hidden zeros and splittings in tree-level scattering amplitudes can be derived from cutting Feynman diagrams. It focuses on ordered amplitudes in Tr(φ³), Yang-Mills, and non-linear sigma model theories. Three types of cuts applied to any diagram divide the amplitude into pieces that match these features: one type for hidden zeros, one for 2-splits, and one for 3-splits called smooth splittings. This provides a diagrammatic explanation independent of an auxiliary orthogonal-space picture. Sympathetic readers would care as it unifies understanding of these amplitude properties across theories.

Core claim

The central claim is that three universal cutting procedures on Feynman diagrams, valid for any diagram, separate a full ordered tree amplitude into two or three pieces corresponding to hidden zeros, 2-splits, and smooth 3-splits in the Tr(φ³), Yang-Mills, and non-linear sigma model theories.

What carries the argument

Three universal ways of cutting Feynman diagrams that separate amplitudes into pieces matching observed zeros and splits.

If this is right

  • The first type of cutting leads to hidden zeros.
  • The second type gives rise to 2-splits.
  • The third type corresponds to 3-splits called smooth splittings.
  • This holds for any diagram in the three theories considered.
  • The results do not depend on the auxiliary orthogonal spaces picture.

Where Pith is reading between the lines

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

  • This cutting approach might extend to loop amplitudes or other theories with similar structures.
  • The cuts could offer a new way to factorize or compute amplitudes by parts.
  • Similar diagrammatic cuts might relate to other known properties like BCJ relations in amplitudes.

Load-bearing premise

The three proposed cutting procedures are valid for every diagram in the three theories and produce pieces that exactly match the hidden zeros and splittings.

What would settle it

A specific Feynman diagram in one of the theories where one of the cutting procedures fails to reproduce the corresponding zero or splitting in the amplitude calculation.

read the original abstract

In this paper, we propose new understandings for recently discovered hidden zeros and novel splittings, by utilizing Feynman diagrams. The study focus on ordered tree level amplitudes of three theories, which are ${\rm Tr}(\phi^3)$, Yang-Mills, and non-linear sigma model. We find three universal ways of cutting Feynman diagrams, which are valid for any diagram, allowing us to separate a full amplitude into two/three pieces. As will be shown, the first type of cuttings leads to hidden zeros, the second one gives rise to $2$-splits, while the third one corresponds to $3$-splits called smooth splittings. Throughout this work, we frequently use the helpful auxiliary technic of thinking the resulting pieces as in orthogonal spaces. However, final results are independent of this auxiliary picture.

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 / 1 minor

Summary. The manuscript claims that hidden zeros and splittings (2-splits and smooth 3-splits) observed in ordered tree-level amplitudes of Tr(φ³), Yang-Mills, and non-linear sigma model can be understood through three universal cutting procedures on Feynman diagrams. These procedures, asserted to apply to any diagram, decompose a full amplitude into two or three pieces, with the first type producing hidden zeros, the second 2-splits, and the third 3-splits; an auxiliary orthogonal-space picture is employed but stated to be non-essential to the final results.

Significance. If the universality holds, the work supplies a uniform, diagram-based account of these amplitude properties across three theories using only standard Feynman manipulations, potentially clarifying their origin without new auxiliary structures.

major comments (2)
  1. [Abstract] Abstract and Introduction: the central assertion that the three cutting procedures are 'valid for any diagram' and exactly reproduce the observed zeros/splittings lacks a general proof or inductive argument covering all topologies, including diagrams with multiple vertices of the same type or higher-valence interactions in NLSM/YM; only illustrative examples are provided, leaving the universality claim load-bearing but unsubstantiated.
  2. [Introduction] The correspondence between each cutting type and the specific zeros or splits is stated without explicit algebraic or diagrammatic definitions of the cutting rules (e.g., no equations specifying momentum routing or vertex selections), making it impossible to verify that the decomposition is exhaustive and free of residual terms for arbitrary diagrams.
minor comments (1)
  1. Notation for the three cutting procedures should be introduced with consistent symbols or labels early in the text to aid readability when referring to them across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We agree that the universality claim and the cutting rules would benefit from more explicit definitions and a general argument, and we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and Introduction: the central assertion that the three cutting procedures are 'valid for any diagram' and exactly reproduce the observed zeros/splittings lacks a general proof or inductive argument covering all topologies, including diagrams with multiple vertices of the same type or higher-valence interactions in NLSM/YM; only illustrative examples are provided, leaving the universality claim load-bearing but unsubstantiated.

    Authors: The manuscript demonstrates the three cutting procedures on representative diagrams of all three theories, including cases with repeated vertices. The procedures are defined locally (selecting a propagator or vertex and routing momenta in a fixed way), so they apply to any topology by construction. We acknowledge that an explicit inductive argument covering arbitrary numbers of vertices and higher-valence cases would make the claim more rigorous. In the revision we will add a short general section proving that the cuts exhaust the amplitude with no residual terms for any diagram. revision: yes

  2. Referee: [Introduction] The correspondence between each cutting type and the specific zeros or splits is stated without explicit algebraic or diagrammatic definitions of the cutting rules (e.g., no equations specifying momentum routing or vertex selections), making it impossible to verify that the decomposition is exhaustive and free of residual terms for arbitrary diagrams.

    Authors: The body of the paper supplies diagrammatic illustrations and verbal descriptions of the three cuts, together with explicit momentum assignments in the examples. We agree that formal equations defining the cuts (momentum routing, vertex selection criteria, and the resulting sub-amplitudes) are needed for verification. The revision will insert these algebraic definitions immediately after the introduction and verify exhaustiveness on a general diagram. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard Feynman-diagram derivation

full rationale

The paper derives three cutting procedures directly from the Feynman rules of Tr(φ³), Yang-Mills and NLSM. These rules are applied to separate amplitudes into pieces that isolate zeros or splits. No quantity is defined in terms of the result it is claimed to produce, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation. The orthogonal-space auxiliary language is explicitly declared non-essential. The derivation therefore remains self-contained against the external benchmark of ordinary Feynman-diagram algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that Feynman diagrams correctly encode the amplitudes of the three theories and on the paper-specific claim that the three cutting operations separate those amplitudes into the stated pieces.

axioms (2)
  • domain assumption Feynman diagrams represent the tree-level amplitudes in Tr(φ³), Yang-Mills, and NLSM
    Standard assumption in perturbative QFT
  • ad hoc to paper The three cutting procedures are valid for any diagram and produce the observed zeros and splits
    Core claim of the work, introduced without derivation in the abstract

pith-pipeline@v0.9.0 · 5654 in / 1402 out tokens · 37806 ms · 2026-05-23T17:46:14.378212+00:00 · methodology

discussion (0)

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Forward citations

Cited by 9 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...

  8. Hidden Zeros and $2$-split via BCFW Recursion Relation

    hep-th 2025-04 unverdicted novelty 4.0

    Hidden zeros in NLSM amplitudes are proven via modified BCFW recursion, with 2-split holding only under careful current definition.

  9. Note on hidden zeros and expansions of tree-level amplitudes

    hep-th 2025-02 unverdicted novelty 4.0

    Hidden zeros in tree-level amplitudes of several theories are attributed to zeros of bi-adjoint scalar amplitudes via universal expansions, with a mechanism shown to cancel potential propagator divergences in gravity.

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