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An Amplituhedron is an infinite-dimensional geometric object which encodes information about scattering amplitudes in a quantum theory.
The first known example applies to the planar sector (i.e. all the Feynman diagrams that are planar graphs) of $ \mathcal N=4 $ Super-Yang-Mills theory, for any number of particles and any distribution of helicities, to all orders in perturbation theory. These classes of Feynman diagrams correspond to regions within the amplituhedron, and the overall amplitude itself is obtained by integrating a volume form over the corresponding region.
Constructing the AmplituhedronEdit this section
It's a geometric object in some space whose dimension depends on the number of external particles, number of loops, and number of "helicity flips". The volume form, the integrand, is a simple form roughly scaling like $ 1/x $ where $ x $ is the distance from a face, and the faces are given by inequalities of the type "determinants of a submatrix are zero". These inequalities depend on the external momenta and/or twistor variables, sort of linearly or simply. The scattering amplitude is the single simple integral of the volume form over the polytope..