Momentum Amplituhedron meets Kinematic Associahedron

TL;DR

This paper explores the deep connection between the momentum amplituhedron and the kinematic associahedron in four-dimensional spacetime, revealing shared singularity structures and form relations in scattering amplitudes of gauge and scalar theories.

Contribution

It establishes a direct link between the canonical forms of the momentum amplituhedron and the associahedron in four dimensions, highlighting their shared singularity and factorization properties.

Findings

The associahedron form sums over all helicity sectors of the reduced momentum amplituhedron.

Shared factorization channels correspond to vanishing Mandelstam variables.

Relations between the canonical forms extend directly on the kinematic space of scalar theory.

Abstract

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.