Causal Diamonds, Cluster Polytopes and Scattering Amplitudes

TL;DR

This paper links cluster polytopes and scattering amplitudes through a (1+1)-dimensional causal structure in kinematic space, revealing geometric and physical insights into amplitude calculations.

Contribution

It identifies a physical origin for cluster polytopes using causal structures and wave equations, connecting them to scattering amplitudes and introducing a new polytope for efficient one-loop amplitude representation.

Findings

Cluster polytopes arise from causal structures in kinematic space.

The polytopes correspond to different scattering amplitude types.

A new polytope simplifies one-loop amplitude calculations.

Abstract

The "amplituhedron" for tree-level scattering amplitudes in the bi-adjoint $\phi^3$ 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.