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arxiv: 2508.21345 · v2 · submitted 2025-08-29 · ✦ hep-th

2-split from Feynman diagrams and Expansions

Bo Feng , Liang Zhang , Kang Zhou This is my paper

Pith reviewed 2026-05-18 21:02 UTC · model grok-4.3

classification ✦ hep-th
keywords 2-splitFeynman diagramsbi-adjoint scalarYang-Millsnonlinear sigma modelgeneral relativitytree-level amplitudesamplitude expansions
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The pith

Tree-level BAS plus Yang-Mills amplitudes split into two independent pieces under specific kinematic conditions due to a recurring pattern in their Feynman vertices.

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

The paper aims to establish that certain tree-level scattering amplitudes split into a product of two simpler amplitudes when the external momenta satisfy a 2-split kinematic condition. The central proof first treats mixed bi-adjoint scalar plus Yang-Mills, nonlinear sigma model, or gravity amplitudes by direct Feynman diagram expansion. The argument hinges on one consistent structural feature appearing in all the relevant interaction vertices. Once the mixed case is settled, the authors insert known expansions that express pure-theory amplitudes as linear combinations of the mixed ones, thereby transferring the split property to the pure Yang-Mills, nonlinear sigma model, and gravity cases. A byproduct is a set of universal formulas that expand the pure-theory currents back into bi-adjoint scalar currents.

Core claim

We prove the 2-split property for tree-level BAS⊕X amplitudes with X=YM,NLSM,GR based on a particular pattern in the Feynman rules of various vertices, and then establish the 2-split behavior for X amplitudes using expansions of X amplitudes into BAS⊕X amplitudes. As a byproduct, we derive universal expansions of the resulting pure X currents into BAS currents, which closely parallel the corresponding on-shell amplitude expansions.

What carries the argument

The recurring pattern in the Feynman rules of the interaction vertices that produces the 2-split when diagrams are summed under the chosen kinematic conditions.

If this is right

  • Pure Yang-Mills tree amplitudes inherit the 2-split property.
  • The same splitting holds for tree amplitudes in the nonlinear sigma model and in general relativity.
  • Universal expansions relate the pure X currents to bi-adjoint scalar currents in a manner parallel to the amplitude expansions.
  • The 2-split supplies a new factorization identity that can be used to simplify higher-multiplicity calculations in these theories.

Where Pith is reading between the lines

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

  • The vertex pattern may reflect an underlying algebraic relation that could be used to generate similar splits in other kinematic regimes.
  • The method supplies an off-shell route to amplitude identities that could be compared with on-shell recursion techniques.
  • If the pattern persists at loop level, analogous 2-split statements might hold for one-loop amplitudes in the same theories.

Load-bearing premise

The proof depends on the existence of one particular structural pattern that appears uniformly across the Feynman rules of the vertices involved.

What would settle it

An explicit calculation of any five-point BAS⊕YM tree amplitude that fails to factor under the 2-split kinematics would falsify the claim.

read the original abstract

In this paper, we investigate the $2$-split behavior of tree-level amplitudes of bi-adjoint scalar (BAS), Yang-Mills (YM), non-linear sigma model (NLSM), and general relativity (GR) theories under certain kinematic conditions. Our approach begins with a proof, based on the Feynman diagram method, of the $2$-split property for tree-level BAS$\oplus$X amplitudes with $\mathrm{X}={\mathrm{YM},\mathrm{NLSM},\mathrm{GR}}$. The proof relies crucially on a particular pattern in the Feynmam rules of various vertices. Building on this, we use the expansion of X amplitudes into BAS$\oplus$X amplitudes to establish the $2$-split behavior. As a byproduct, we derive universal expansions of the resulting pure X currents into BAS currents, which closely parallel the corresponding on-shell amplitude expansions.

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 / 1 minor

Summary. The paper claims to prove the 2-split property for tree-level BAS⊕X amplitudes (X = YM, NLSM, GR) via the Feynman diagram method, relying on a particular pattern in the vertex Feynman rules. It then uses expansions of X amplitudes into BAS⊕X amplitudes to establish the 2-split for pure X amplitudes, and as a byproduct derives universal expansions of the resulting pure X currents into BAS currents that parallel known on-shell amplitude expansions.

Significance. If the central claims hold, the work would supply a diagrammatic foundation for the 2-split property across these theories and furnish explicit current expansions relating BAS to the other models. These results could streamline calculations and reveal structural relations among amplitudes in different theories.

major comments (1)
  1. [Proof of 2-split for BAS⊕X amplitudes] The proof that tree-level BAS⊕X amplitudes obey 2-split (the step that enables the subsequent expansion argument for pure X) rests on invoking a 'particular pattern' in the Feynman rules of the various vertices. No separate lemma or derivation is supplied that extracts this pattern directly from the Lagrangian definitions and demonstrates that it holds identically for every contributing diagram and for arbitrary multiplicity n under the stated kinematic conditions. This pattern is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] Abstract: 'Feynmam rules' is a typographical error and should read 'Feynman rules'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point where our presentation of the proof can be strengthened. We agree that the 'particular pattern' in the vertex rules is central to the argument and will add an explicit derivation in the revision.

read point-by-point responses

  1. Referee: [Proof of 2-split for BAS⊕X amplitudes] The proof that tree-level BAS⊕X amplitudes obey 2-split (the step that enables the subsequent expansion argument for pure X) rests on invoking a 'particular pattern' in the Feynman rules of the various vertices. No separate lemma or derivation is supplied that extracts this pattern directly from the Lagrangian definitions and demonstrates that it holds identically for every contributing diagram and for arbitrary multiplicity n under the stated kinematic conditions. This pattern is load-bearing for the central claim.

    Authors: We acknowledge the validity of this observation. The pattern is extracted from the standard Feynman rules obtained from the respective Lagrangians, but we agree that a self-contained derivation is desirable. In the revised version we will insert a new subsection (prior to the main proof) that: (i) recalls the relevant interaction vertices for BAS, YM, NLSM and GR; (ii) states the kinematic conditions; and (iii) proves by direct inspection of the color and kinematic factors that the stated pattern holds for every tree diagram at arbitrary multiplicity. This will be presented as a lemma whose hypotheses are satisfied identically by the Feynman rules of each theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from Feynman diagram summation and internally derived expansions

full rationale

The paper establishes the 2-split for BAS⊕X via explicit Feynman diagram analysis relying on a vertex rule pattern, then applies expansions of X amplitudes into BAS⊕X that are derived as a byproduct within the same work. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims retain independent content from the diagram method and expansion construction. The argument is self-contained against the stated kinematic conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on an assumed pattern in Feynman rules for the mixed vertices and on the validity of the cited expansions that relate X amplitudes to BAS⊕X amplitudes.

axioms (1)
  • domain assumption Particular pattern in the Feynman rules of various vertices
    Abstract states the proof relies crucially on this pattern.

pith-pipeline@v0.9.0 · 5668 in / 1170 out tokens · 40702 ms · 2026-05-18T21:02:15.278711+00:00 · methodology

discussion (0)

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Forward citations

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  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

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  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. 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.

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