Causal Diamonds, Cluster Polytopes and Scattering Amplitudes

TL;DR

The paper links scattering amplitudes to positive geometries realized as polytopes by embedding kinematic space in a (1+1)-dimensional causal setting and solving a positivity-constrained wave equation. Time evolution is recast as mutations on Dynkin quivers, yielding finite cluster polytopes that reproduce ABHY associahedra for type A and extend to B, C, and D types, connecting to tree- and one-loop amplitudes. It provides explicit constructions for D_n and a halved bar D_n to efficiently capture tadpole diagrams, with factorization properties governed by removing nodes from Dynkin diagrams and recursion relations derived from polytope projections. The framework offers a geometrically natural, recursion-friendly route to amplitudes, with potential extensions to higher loops, exceptional cluster polytopes, and stringy generalizations. Overall, it reveals a deep, positive-geometry structure underlying tree and one-loop scattering that mechanizes factorization and suggests new computational strategies.

Abstract

The "amplituhedron" for tree-level scattering amplitudes in the bi-adjoint $φ^3$ theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1+1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic "spacetime" with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain "walk", associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The ${\cal A}_{n{-}3},{\cal B}_{n{-}1}/{\cal C}_{n{-}1}$ and ${\cal D}_n$ polytopes are the amplituhedra for $n$-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope $\bar{\cal D}_n$, which chops the ${\cal D}_n$ polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

Figures (20)

• Figure 1: The correspondence between cluster algebras, Dynkin quivers and scattering amplitudes
• Figure 2: The infinite array of planar variables.
• Figure 3: Diagonals on an n-gon and the corresponding points in the $(1+1)$-dimensional space-time.
• Figure 4: Examples of minimal non-redundant regions.
• Figure 5: A causal diamond is associated to a "Gauss law" involving variables sitting at its corners