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

TL;DR

This paper provides a one-loop derivation of the duality between Wilson polygons and MHV amplitudes, revealing its origin through variable changes and dimensional relations, and explores its generalizations and geometric interpretations.

Contribution

It presents a novel proof of the Wilson polygon - MHV amplitude duality at one-loop level using Feynman parametrization and dimensional relations, and introduces a generalization involving vertex operators.

Findings

Duality proven at one-loop level via variable change

Generalization with vertex operator insertion for 3-point functions

Discussion of Landau equations and hyperbolic geometry interpretations

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.