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arxiv: 2507.14583 · v2 · submitted 2025-07-19 · ✦ hep-th

BCFW like recursion for Deformed Associahedron

Sujoy Mahato , Sourav Roychowdhury This is my paper

Pith reviewed 2026-05-19 04:16 UTC · model grok-4.3

classification ✦ hep-th
keywords BCFW recursiondeformed associahedronABHY associahedroncluster polytopesmulti-scalar cubic theoryscattering amplitudespositive geometryprojective triangulation
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The pith

BCFW-like recursions apply to the deformed ABHY associahedron and compute amplitudes in multi-scalar cubic theories.

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

The paper adapts BCFW-style recursion relations to a deformed realization of the ABHY associahedron that lives in kinematic space. This geometry encodes tree-level scattering amplitudes for theories containing multiple distinct scalar particles whose cubic interactions have unequal coupling strengths. The recursion proceeds by summing contributions that match the projective triangulations of the deformed polytope. The same approach extends to deformed D-type cluster polytopes, which capture the corresponding one-loop amplitudes. Readers interested in geometric methods for amplitude calculation may care because the recursion replaces diagram-by-diagram summation with a single geometric decomposition.

Core claim

The paper establishes that BCFW-like recursion relations apply to the deformed realization of the ABHY-associahedron in kinematic space for a theory with multiple scalar particles and cubic couplings of different strengths. The recursion terms correspond to projective triangulations of this geometry. The formalism generalizes directly to the deformed realization of D-type cluster polytopes that encode one-loop amplitudes in the same class of cubic theories. The paper also notes the possibility of recovering effective-field-theory amplitudes from the underlying cubic theory through these recursion relations.

What carries the argument

The deformed ABHY-associahedron (and its D-type cluster-polytope generalizations), a positive geometry in kinematic space whose canonical form encodes the amplitudes; recursion is realized by deforming the standard BCFW shift so that each term corresponds to a projective triangulation.

If this is right

  • Tree-level amplitudes are obtained by summing the recursive contributions that arise from projective triangulations of the deformed associahedron.
  • One-loop amplitudes in the same theories follow from the analogous recursion on deformed D-type cluster polytopes.
  • Projective triangulations supply a systematic geometric decomposition of each amplitude.
  • Limits or modifications of the recursion may isolate effective-field-theory corrections to the cubic amplitudes.

Where Pith is reading between the lines

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

  • The same deformation-plus-recursion pattern may extend to other families of cluster polytopes beyond the A- and D-type cases.
  • Numerical verification on low-multiplicity amplitudes with concrete deformation parameters would provide an immediate consistency check.
  • The geometric recursion could illuminate amplitude structures when couplings break flavor symmetries or introduce mass scales.

Load-bearing premise

The deformed ABHY-associahedron correctly encodes the tree-level scattering amplitudes of the multi-scalar cubic theory with unequal coupling strengths.

What would settle it

Explicitly compute a four- or five-point amplitude with two distinct scalar species and unequal cubic couplings using the proposed recursion on the deformed associahedron, then compare the result term-by-term to the sum of all Feynman diagrams; any mismatch falsifies the claim.

Figures

Figures reproduced from arXiv: 2507.14583 by Sourav Roychowdhury, Sujoy Mahato.

Figure 1
Figure 1. Figure 1: The projectection of 1d associahedron on s-axis via recursion In our case m = 1 and the deformed 1d associahedron is located in the s-t plane by the equation αs + βt = c. Hence for our case the volume V in (5.1) takes the form Vt = ⟨Z⋆Z1⟩ ⟨Y Z⋆⟩⟨Y Z1⟩ = α  1 αs + 1 βt . (5.4) So it seems we need to project along the proper direction to match the geometric picture with recursion term. 5.2 Six point massle… view at source ↗
Figure 2
Figure 2. Figure 2: Prism formed by projecting X36 facet onto X13X14 edge We will now illustrate how we assign coordinates to the vertices of the prism. Let us choose the basis to be {X13,X14,X15}. Let us consider the vertex Z3. The coordinate for Z3 := {X36,X46,A} is obtained by solving the following equations : X36 = 1 α36  − α14X14 + c36 = 0 , 20 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Prism formed by projecting X2¯3 facet onto Y1Y2 edge We give a similar geometric interpretation of the recursion terms for the loop case. As we have already seen there are six terms in the recursion corresponding to (Y1,Y2) rescaling. Among 22 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two distinct deformed realization of the associahedron in ( [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The edges X35 and X24 onto the X14 edge via recursion. Z⋆ =  1, c24 α13 ,0  ; Z1 =  1, c25 α13 ,0  ; Z2 =  1, c25 α13 , c25 − c24 α14  ; Y =  1,X13,X14 . (A.3) Hence the volume will be VT =  ⟨Z⋆Z1Z2⟩ 2 ⟨Y Z⋆Z1⟩⟨Y Z1Z2⟩⟨Y Z⋆Z2⟩ =  α13α14c25 − c24 α14X14 c24 − α13X13 + α14X14c25 − α13X13 . (A.4) Now for the rectangle part we break it into two triangle with VT1 and VT2 . Hence the volume of … view at source ↗
read the original abstract

In this paper, we explore the applicability of the BCFW-like recursion relations \cite{He:2018svj,Yang:2019esm} to a wider class of positive geometries. Previously it was found in \cite{Jagadale:2022rbl}, the tree level scattering amplitude of a theory with more than one type of scalar particles interacting via cubic couplings of different strength can be captured by a deformed realization of the ABHY-associahedorn in the kinematic space. In the literature, we explore the adaptation of the recursion relations for the case of deformed associahedron. The formalism is further generalized to the deformed realization of the D-type cluster polytopes which captures the one-loop amplitudes in this class of cubic theories. These recursion terms correspond to projective triangulation of the associahedron (or D-type cluster polytopes). Towards the end, we briefly mention the idea of recovering EFT amplitudes from the cubic theory in terms of recursion relations.

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

0 major / 3 minor

Summary. The manuscript adapts BCFW-like recursion relations to the deformed realization of the ABHY-associahedron in kinematic space, which encodes tree-level amplitudes for multi-scalar cubic theories with varying coupling strengths. It identifies the recursion terms with projective triangulations and extends the construction to deformed D-type cluster polytopes to capture one-loop amplitudes in the same class of theories. The work concludes with a brief discussion of recovering EFT amplitudes from the underlying cubic theory via these recursions, building directly on prior results for the deformed geometries.

Significance. If the formal adaptation holds, the result extends the reach of positive-geometry recursions to deformed associahedra and cluster polytopes, offering a structured way to compute amplitudes in theories with multiple scalar species and non-uniform cubic couplings. The manuscript correctly credits the prior literature for the amplitude-encoding property of the deformed geometries and focuses on showing that the recursion proceeds in the same manner. No new machine-checked proofs, reproducible code, or independent falsifiable predictions are provided, so the primary contribution is the generalization itself rather than a standalone computational advance.

minor comments (3)
  1. The abstract and introduction cite the deformed ABHY-associahedron from prior work but do not restate the precise kinematic deformation map; including a short recap of the deformed coordinates (e.g., the modified Mandelstam variables) would improve readability for readers unfamiliar with the cited reference.
  2. In the section describing the generalization to D-type cluster polytopes, the correspondence between recursion terms and projective triangulations is stated formally; an explicit low-point example (such as the one-loop 4-point case) would make the identification concrete and help verify that the deformation preserves the required pole factorization.
  3. The brief discussion of EFT amplitude recovery at the end would benefit from a clearer statement of which higher-dimension operators are generated and how the recursion selects them, perhaps with a reference to the relevant effective Lagrangian terms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. The provided summary accurately reflects the scope of our work, which generalizes BCFW-style recursions to deformed ABHY associahedra and D-type cluster polytopes in the context of multi-scalar cubic theories. We appreciate the recognition that the contribution lies in this formal extension building on prior results for the deformed geometries.

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper cites prior work [Jagadale:2022rbl] for the fact that the deformed ABHY-associahedron encodes tree-level amplitudes in multi-scalar cubic theories, then adapts the independently established BCFW recursion relations from He:2018svj and Yang:2019esm to this geometry. The identification of recursion terms with projective triangulations and the extension to deformed D-type cluster polytopes for one-loop amplitudes constitute a formal mapping rather than any definitional equivalence or reduction of the output to the input by construction. No equations in the provided text show a fitted parameter renamed as a prediction, a self-referential definition, or a load-bearing uniqueness theorem imported from the same authors. The derivation remains self-contained as an application of known recursion techniques to a previously defined positive geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the central claims rest on domain assumptions imported from earlier papers about the geometric encoding of amplitudes and the applicability of recursion to positive geometries; no free parameters or new invented entities are explicitly introduced in the summary.

axioms (2)
  • domain assumption The deformed realization of the ABHY-associahedron in kinematic space captures tree-level scattering amplitudes of multi-scalar cubic theories with different coupling strengths.
    Invoked via citation to Jagadale:2022rbl as the starting point for the recursion adaptation.
  • domain assumption BCFW-like recursion relations can be applied to positive geometries beyond the standard associahedron.
    Taken from the cited works He:2018svj and Yang:2019esm.

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Reference graph

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