One-loop derivation of the Wilson polygon - MHV amplitude duality

TL;DR

This work establishes a concrete one-loop mapping between Wilson polygons and MHV amplitudes by a targeted change of Feynman-parameter variables and a dimension-shifting relation that connects $D=6$ and $D=4$ scalar box integrals. The key result is that the finite part of the $D=4$ two-mass-easy box corresponds to a Wilson-polygon quantity, enabling a precise duality for one-loop MHV amplitudes; the construction leverages Tarasov–Nizic dimensional relations and unitarity (via generalized cuts) to extend the link to Wilson-loop observables and their imaginary parts. The analysis also explores the 3-point case with a Wilson triangle vertex operator and develops the Landau-equation framework and hyperbolic-geometry interpretation for one-loop diagrams, connecting UV/IR structure to cusp dynamics in AdS-like spaces and Liouville/2D YM perspectives. The discussion points to prospects for NMHV extensions, higher-loop generalizations, and deeper ties to moduli spaces and knot theory. Overall, the paper provides a geometrically flavored, perturbative bridge between Wilson-loop constructs and scattering amplitudes in ${ m \\mathcal{N}=4}$ SYM at one loop.

Abstract

We discuss the origin of the Wilson polygon - MHV amplitude duality at the perturbative level. It is shown that the duality for the MHV amplitudes at one-loop level can be proven upon the peculiar change of variables in Feynman parametrization and the use of the relation between Feynman integrals at the different space-time dimensions. Some generalization of the duality which implies the insertion of the particular vertex operator at the Wilson triangle is found for the 3-point function. We discuss analytical structure of Wilson loop diagrams and present the corresponding Landau equations. The geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed.