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 \( \mathcal{A} \)n−3, \( \mathcal{B} \)n−1/\( \mathcal{C} \)n−1 and \( \mathcal{D} \)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 \( \overline{\mathcal{D}} \)n, which chops the \( \mathcal{D} \)n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.
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Arkani-Hamed, N., He, S., Salvatori, G. et al. Causal diamonds, cluster polytopes and scattering amplitudes. J. High Energ. Phys. 2022, 49 (2022). https://doi.org/10.1007/JHEP11(2022)049
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DOI: https://doi.org/10.1007/JHEP11(2022)049