From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
Arthur E. Lipstein, Lionel Mason
TL;DR
This work shows that the planar S-matrix for N=4 SYM can be computed from a holomorphic Wilson loop in momentum twistor space, with loop integrands that are locally in dlog form. The authors derive Feynman rules in this twistor framework, demonstrate how dlog structures arise at all loop orders for MHV amplitudes, and illustrate the method through explicit 1-loop and 2-loop MHV cases and 1-loop NMHV. Central to the approach are contour choices tied to reality conditions, which encode external data and determine the integral evaluation. The results point toward a general mechanism by which all-loop MHV amplitudes exhibit hidden simplicity via dlog differential forms, with potential connections to Grassmannian formulations and future avenues for systematic evaluation.
Abstract
The S-matrix for planar N = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand via MHV diagrams in twistor space. We show that at MHV, this formulation leads directly to expressions for loop integrands in d log form; i.e., the integrand is a product of exterior derivatives of logarithms of rational functions. For higher MHV degree, it is in d log form multiplied by delta functions. The parameters appearing in the d log form arise geometrically as the coordinates of insertion points of propagators on the holomorphic Wilson loop or on MHV vertices. We discuss a number of examples at one and two loops and give a preliminary discussion of the evaluation of the 1-loop MHV amplitude.