The twistor Wilson loop and the amplituhedron

TL;DR

This paper investigates the geometric relationship between the twistor Wilson loop and the amplituhedron in N=4 SYM, revealing that beyond NMHV amplitudes, Feynman diagram images do not tessellate the amplituhedron without unmatched boundaries.

Contribution

It demonstrates that the geometric images of Wilson loop Feynman diagrams cannot form a perfect tessellation of the amplituhedron beyond NMHV amplitudes.

Findings

Feynman diagram images have physical and spurious boundaries.

Spurious boundaries should match between diagrams.

No geometric image of Wilson loop diagrams can avoid unmatched spurious boundaries beyond NMHV.

Abstract

The amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We prove however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be "internal", matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving unmatched spurious boundaries.