Notes on the scattering amplitude / Wilson loop duality

TL;DR

This paper explores the duality between scattering amplitudes and Wilson loops in supersymmetric Yang-Mills theory, extending it to all helicity states and loop levels using recursion relations and supersymmetric Wilson loops.

Contribution

It introduces a supersymmetric extension of Wilson loops and demonstrates the duality's validity at all perturbative orders through recursion relations.

Findings

Recursion relations reproduce BCFW amplitudes

Duality holds at loop level for all helicity states

Finite correlation functions can compute derivatives of MHV amplitudes

Abstract

We consider the duality between the four-dimensional S-matrix of planar maximally supersymmetric Yang-Mills theory and the expectation value of polygonal shaped Wilson loops in the same theory. We extend the duality to amplitudes with arbitrary helicity states by introducing a suitable supersymmetric extension of the Wilson loop. We show that this object is determined by a host of recursion relations, which are valid at tree level and at loop level for a certain "loop integrand" defined within the Lagrangian insertion procedure. These recursion relations reproduce the BCFW ones obeyed by tree-level scattering amplitudes, as well as their extension to loop integrands which appeared recently in the literature, establishing the duality to all orders in perturbation theory. Finally, we propose that a certain set of finite correlation functions can be used to compute all first derivatives of the logarithm of MHV amplitudes.