Notes on the scattering amplitude / Wilson loop duality

TL;DR

This work establishes a robust duality between planar N=4 SYM scattering amplitudes and a supersymmetric Wilson loop, using momentum twistsors to reveal dual conformal structure and Lagrangian-insertion techniques to define loop integrands. It demonstrates that tree-level amplitudes are captured by a super-Wilson loop whose recursion relations match the BCFW framework, and that the same formalism extends to loop integrands, yielding equivalence to all orders at the integrand level. The construction clarifies the role of the self-dual sector in tree amplitudes, connects to Mason–Skinner's superconnection, and provides a practical framework for computing derivatives of logarithms of MHV amplitudes via finite correlation functions. Overall, the paper solidifies the amplitude-Wilson loop duality and offers a unified, symmetry-guided approach to perturbative N=4 SYM across all loops.

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.