Abstract
The momentum amplituhedron is a positive geometry encoding tree-level scattering amplitudes in mathcal{N} = 4 super Yang-Mills directly in spinor-helicity space. In this paper we classify all boundaries of the momentum amplituhedron ℳn, k and explain how these boundaries are related to the expected factorization channels, and soft and collinear limits of tree amplitudes. Conversely, all physical singularities of tree amplitudes are encoded in this boundary stratification. Finally, we find that the momentum amplituhedron ℳn, k has Euler characteristic equal to one, which provides a first step towards proving that it is homeomorphic to a ball.
Highlights
JHEP07(2020)201 many properties with the hypersimplex ∆k+1,n, which is a well studied convex polytope with known boundary structure
We show how the physical singularities of the amplitude are encoded in the boundaries of this geometry
We provide evidence that, for the extensive number of cases analysed, this positive geometry has Euler characteristic one which suggests that the momentum amplituhedron is homeomorphic to a ball and it is simpler than the ordinary amplituhedron
Summary
Let us begin by reviewing the well-known behaviour of colour-ordered amplitudes under collinear and soft limits at tree level. These are governed by universal functions, which depend only on the particles which become collinear, or, in the case of soft limits, on the nearest neighbours of the soft particle. We provide an explicit form of the super-splitting functions for the case when two super-particles become collinear. One finds two such functions, which correspond to a helicity-preserving and a helicity-decreasing case. One finds two super-soft limits and each of them can be thought of as two simultaneous collinear limits of the same type
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