From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes

TL;DR

This paper demonstrates that planar N=4 super Yang-Mills loop integrands can be expressed in d log form using twistor space Wilson loops, revealing geometric structures and simplifying amplitude calculations.

Contribution

It introduces a direct derivation of loop integrands in d log form from holomorphic Wilson loops in twistor space, extending to higher MHV degrees with delta function factors.

Findings

At MHV, integrands are in d log form.

Higher MHV degrees involve delta functions.

Examples at one and two loops illustrate the approach.

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.