Scattering amplitudes from the canonical form of the amplituhedron

Show that in planar N=4 supersymmetric Yang–Mills theory, the n-point tree-level scattering amplitude equals the canonical form of the amplituhedron A_{k,n,m}(Z), and that the L-loop scattering amplitude equals the integral of an appropriate regulator function against the canonical form of the loop amplituhedron as defined by Arkani-Hamed and Trnka.

Background

The amplituhedron framework proposes a geometric reformulation of scattering amplitudes in planar N=4 SYM, replacing sums over Feynman diagrams with canonical forms on positive geometries. At tree level, the conjecture posits an exact identification between the amplitude and the amplituhedron’s canonical form; at loop level, an integral over a regulator function against the canonical form of a loop amplituhedron is expected to reproduce the amplitude.

A proof would rigorously establish the amplituhedron as a complete geometric model for amplitudes, strengthening the link between positive geometry and quantum field theory computations.