Abstract
In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in mathcal{N} = 4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.
Highlights
Despite the name, the amplituhedron is more naturally suited to describe the dual Wilson loop rather than the amplitude itself, being defined in the momentum twistor space
After imposing additional assumptions on Λ and Λ, which will guarantee positive planar Mandelstam variables, we claim that the momentum amplituhedron Mn,k is a positive geometry and its volume form encodes the n-particle Nk−2MHV tree-level scattering amplitude in N = 4 SYM
In this paper we have introduced a novel geometric object, the momentum amplituhedron Mn,k, which computes tree-level scattering amplitudes in N = 4 SYM directly in momentum space
Summary
We start by recalling the construction of the amplituhedron in momentum twistor space. The original construction of the amplituhedron defines the volume form Ω(nm,k) as differential form on an auxiliary Grassmannian space G(k , k +m) parametrized by Y. It can be accomplished by replacing the differential with respect to Y by the differential with respect to the kinematic data Z: dY log → dZlog, and at the same time by fixing Y = Y ∗, where Y ∗ is a reference k -plane in k + m dimensions This new differential form is a logarithmic differential form on the space of configurations of zia, satisfying particular sign-flip or topological conditions [15]. The sign-flip characterization of the amplituhedron does not refer either to any auxiliary space or the quite peculiar bosonization described above, it is not an easy task to find the volume form directly from this definition. We often refer back to the original construction of the amplituhedron in the bosonized space
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