Causal diamonds, cluster polytopes and scattering amplitudes

Abstract

Abstract The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint ϕ3 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 $$ \mathcal{A} $$ A n−3,$$ \mathcal{B} $$ B n−1/$$ \mathcal{C} $$ C n−1 and $$ \mathcal{D} $$ 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 $$ \overline{\mathcal{D}} $$ D ¯ n, which chops the $$ \mathcal{D} $$ D n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.