[Paper Review] Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes

Qinglin Yang|arXiv (Cornell University)|Dec 19, 2019

Advanced Combinatorial Mathematics23 references4 citations

TL;DR

This paper introduces a one-parameter deformation of kinematic variables to derive new recursion relations for tree and one-loop amplitudes in bi-adjoint $φ^3$ theory, generalizing previous BCFW-like formulas. The method geometrically triangulates generalized ABHY associahedra—such as type $Π$, $Π$, and $Π$ polytopes—by projecting onto facets, yielding efficient recursive computations of canonical forms, including novel triangulations beyond soft- and forward-limit cases.

ABSTRACT

In this note we make a field-theoretical derivation of a series of new recursion relations by a one-parameter deformation of kinematic variables for tree and one-loop amplitudes of bi-adjoint $ϕ^3$ theory. Tree amplitudes are given by canonical forms/functions of associahedra realized in kinematic space by Arkani-Hamed, Bai, He and Yan (ABHY); the construction has been extended to generalized associahedra, where type $\mathcal{B}$/$\mathcal{C}$ polytopes compute tadpole diagrams and type $\mathcal{D}$ polytopes compute one-loop planar $ϕ^3$ amplitudes. The new recursions are natural generalizations of the formula we found in 1810.08508, and are shown to work for all "C-independent" ABHY polytopes. Geometrically, the formula indicates triangulation of the generalized associahedron by projecting the whole polytope onto its boundary determined by the deformation. When projecting onto one facet, our recursion gives, e.g. "soft-limit triangulation" and "forward-limit triangulation" for tree and one-loop level. But we also find a lot of new formulae from our recursion relation, by projecting onto lower dimensional facets.

Motivation & Objective

• To develop a field-theoretical derivation of recursion relations for tree and one-loop amplitudes in bi-adjoint $φ^3$ theory.
• To generalize existing BCFW-like recursion formulas by introducing a one-parameter deformation of kinematic variables.
• To establish a geometric framework for triangulating generalized ABHY associahedra (type $Π$, $Π$, $Π$) via projection onto facets.
• To identify and compute new classes of triangulations beyond soft-limit and forward-limit types, including those from lower-dimensional facets.
• To demonstrate that the recursion applies universally to all 'C-independent' ABHY polytopes, ensuring broad applicability across combinatorial types.

Proposed method

• A one-parameter deformation is applied to kinematic variables $X_{ij}$, enabling recursive decomposition of amplitudes via projection of the full polytope onto its boundary.
• The recursion is constructed by projecting the generalized associahedron onto a specific facet, inducing a triangulation where each term corresponds to a canonical form of a prism-like cell.
• The method generalizes earlier 'soft-limit' and 'forward-limit' triangulations by allowing projection onto any facet, not just maximal ones.
• The recursion is applied to type $Π$ (associahedra), $Π$ (cyclohedra), and $Π$ (one-loop amplituhedra), showing consistency across different polytope types.
• Canonical forms of the resulting prismatic cells are computed as products of the bottom cell’s form and the height, enabling recursive evaluation.
• Explicit expressions for 1-loop amplitudes (e.g., $A^{1\text{-}loop}_4$) are derived using this method, with terms decomposed into rational functions of planar variables $X_A$.

Experimental results

Research questions

• RQ1 Can a unified recursion relation be derived for tree and one-loop amplitudes in bi-adjoint $φ^3$ theory using a kinematic deformation?
• RQ2 How does projecting the ABHY polytope onto different facets—especially lower-dimensional ones—generate new triangulations of generalized associahedra?
• RQ3 What is the geometric and algebraic structure of the resulting recursion when applied to type $Π$, $Π$, and $Π$ polytopes?
• RQ4 Can the recursion be generalized beyond soft- and forward-limit cases to include new classes of triangulations?
• RQ5 What is the role of 'C-independence' in ensuring the recursion's universality across different ABHY polytope types?

Key findings

• The recursion successfully computes tree and one-loop amplitudes for all 'C-independent' ABHY polytopes, including type $Π$, $Π$, and $Π$.
• Projection onto a facet yields a triangulation where each term is a product of the canonical form of the base and the height, generalizing soft- and forward-limit cases.
• The method produces new triangulation formulae not previously known, especially when projecting onto lower-dimensional facets.
• Explicit results for the 1-loop 4-point amplitude $A^{1\text{-}loop}_4$ are derived, with five terms ($Ω_3, Ω_{31}, Ω_{23}, Ω_{34}, Ω_{42}$) expressed as rational functions of planar variables $X_A$.
• The recursion is shown to be geometrically consistent with the additive property of canonical forms under triangulation, confirming its validity across different polytope types.
• Numerical checks confirm that the derived amplitudes match known results, validating the recursion’s correctness and efficiency.