Internal boundaries of the loop amplituhedron

TL;DR

This paper reveals internal boundaries within the loop amplituhedron in planar N=4 super Yang-Mills theory, challenging the traditional positive geometry framework and proposing a generalized concept to account for these features.

Contribution

It introduces the notion of internal boundaries in the loop amplituhedron, extending positive geometry to include such features and proposing weighted positive geometries.

Findings

Identification of internal boundaries in the loop amplituhedron

Generalization of positive geometry to include internal boundaries

New all-order residues for the deepest cut of N=4 amplitudes

Abstract

The strict definition of positive geometry implies that all maximal residues of its canonical form are $\pm 1$. We observe, however, that the loop integrand of the amplitude in planar $\mathcal{N}=4$ super Yang-Mills has maximal residues not equal to $\pm 1$. We find the reason for this is that deep in the boundary structure of the loop amplituhedron there are geometries which contain internal boundaries: codimension one defects separating two regions of opposite orientation. This phenomenon requires a generalisation of the concept of positive geometry and canonical form to include such internal boundaries and also suggests the utility of a further generalisation to `weighted positive geometries'. We re-examine the deepest cut of $\mathcal{N}=4$ amplitudes in light of this and obtain new all order residues.