SciPost logo

Internal boundaries of the loop amplituhedron

Gabriele Dian, Paul Heslop, Alastair Stewart

SciPost Phys. 15, 098 (2023) · published 18 September 2023


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

Cited by 3

Crossref Cited-by

Authors / Affiliation: mappings to Contributors and Organizations

See all Organizations.
Funders for the research work leading to this publication