Abstract
In this paper we study a relation between two positive geometries: the momen- tum 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 φ3 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 forms 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 sin- gularity 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.
Highlights
Can be written as integrals over the moduli space of Riemann spheres [1, 2]
In this paper we study a relation between two positive geometries: the momentum amplituhedron, relevant for tree-level scattering amplitudes in N = 4 super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar φ3 theory
The main result of this paper is that these forms are related to each other!1 More precisely, when we evaluate the associahedron form on the little group invariant space, it is the same as the sum of reduced momentum amplituhedron forms summed over all helicity sectors
Summary
We begin by reviewing the two positive geometries which will be considered in this paper: the momentum amplituhedron [7] and the kinematic associahedron [8]. Ometry comes equipped with a canonical differential form, whose leading singularities or residues on zero-dimensional boundaries equal ±1. The canonical differential form of the momentum amplituhedron, respectively associahedron, encodes the tree-level scattering amplitudes for N = 4 sYM, respectively bi-adjoint φ3, theory. In both cases, we provide their definitions in their respective kinematic space in terms of the intersection of a positive region with an affine subspace. For an extensive review on these and other positive geometries, see [10]
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