Generalized permutohedra in the kinematic space

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 $e_1-e_2,\ldots, e_{n-2}-e_{n-1}$.

Figures (7)

• Figure 1: A multi-peripheral Feynman diagram, corresponding to the polyhedral cone defined by Equation \ref{['eqn:kinematicPlate']}. Here $\sigma=(\sigma_1,\ldots, \sigma_n)$ is any permutation of $\{1,\ldots, n\}$.
• Figure 2: Zonotopal generalized permutohedron for Example \ref{['example: zonotopal generalized permutohedron']}, $(c_{12},c_{23},c_{13})=(2,1,3)$. All edges parallel to $e_i-e_j$ have length $c_{ij}$.
• Figure 3: Exploding a point to a hexagon: $(c_{12},c_{13},c_{23})=(\varepsilon,\varepsilon,\varepsilon)$ for $\varepsilon\ge 0$
• Figure 4:
• Figure 5:

Theorems & Definitions (7)

• proof
• proof
• proof : Sketch of proof
• proof
• proof
• proof
• proof