Generalized permutohedra in the kinematic space

TL;DR

This paper explores the geometry of generalized permutohedra within the kinematic space, linking polyhedral structures to differential forms and scattering amplitudes, and providing explicit realizations and interpretations of these geometric objects.

Contribution

It explicitly characterizes the family of zonotopal generalized permutohedra in the kinematic space and interprets scalar amplitude formulas as sums over boundary components of root cones.

Findings

The family of permutohedral polyhedra fills the configuration space of zonotopal generalized permutohedra.

The poles of a certain differential form determine a permutohedral family with the same face lattice.

Mizera's formula for biadjoint scalar amplitude can be interpreted as a sum over boundary components of a root cone.

Abstract

In this note, we study the permutohedral geometry of the poles of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions $e_i-e_j$. Finally we interpret Mizera's formula for the biadjoint scalar amplitude $m(\mathbb{I}_n,\mathbb{I}_n)$, restricted to a certain dimension $n-2$ subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots $e_1-e_2,\ldots, e_{n-2}-e_{n-1}$.