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In this paper we study a relation between two positive geometries: the momentum amplituhedron, relevant for tree-level scattering amplitudes in $mathcal{N} = 4$ super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar $phi^3$ theory. We study the implications of restricting the latter to four spacetime dimensions and give a direct link between its canonical form and the canonical form for the momentum amplituhedron. After removing the little group scaling dependence of the gauge theory, we find that we can compare the resulting reduced form with the pull-back of the associahedron form. In particular, the associahedron form is the sum over all helicity sectors of the reduced momentum amplituhedron forms. This relation highlights the common singularity structure of the respective amplitudes; in particular the factorization channels, corresponding to vanishing planar Mandelstam variables, are the same. Additionally, we also find a relation between these canonical forms directly on the kinematic space of the scalar theory when reduced to four spacetime dimensions by Gram determinant constraints. As a by-product of our work we provide a detailed analysis of the kinematic spaces relevant for the four-dimensional gauge and scalar theories, and provide direct links between them.
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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 $mathcal{M}_{n,k}$ 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.
In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.
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 $mathcal{M}_{n,k}$ 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 $mathcal{M}_{n,k}$ has Euler characteristic equal to one, which provides a first step towards proving that it is homeomorphic to a ball.
In this paper we provide a formula for the canonical differential form of the hypersimplex $Delta_{k,n}$ for all $n$ and $k$. We also study the generalization of the momentum amplituhedron $mathcal{M}_{n,k}$ to $m=2$, and we conclude that the existing definition does not possess the desired properties. Nevertheless, we find interesting momentum amplituhedron-like logarithmic differential forms in the $m=2$ version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms.
Positive geometries provide a modern approach for computing scattering amplitudes in a variety of physical models. In order to facilitate the exploration of these new geometric methods, we introduce a Mathematica package called ``amplituhedronBoundaries for calculating the boundary structures of three positive geometries: the amplituhedron $mathcal{A}_{n,k}^{(m)}$, the momentum amplituhedron $mathcal{M}_{n,k}^{(m)}$ and the hypersimplex $Delta_{k,n}$. The first two geometries are relevant for scattering amplitudes in planar $mathcal{N}=4$ SYM, while the last one is a well-studied polytope appearing in many contexts in mathematics, and is closely related to $mathcal{M}_{n,k}^{(2)}$. The package includes an array of useful tools for the study of these positive geometries, including their boundary stratifications, drawing their boundary posets, and additional tools for manipulating combinatorial structures useful for positive Grassmannians.
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