Internal boundaries of the loop amplituhedron

Abstract

The strict definition of positive geometry implies that all maximal residues of its canonical form are ±1±1. We observe, however, that the loop integrand of the amplitude in planar N=4 super Yang-Mills has maximal residues not equal to ± 1±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 N=4 amplitudes in light of this and obtain new all order residues.