Light-like Wilson loops and the $ar{Q}$-equation

In recent work we began a study of the correlators of multiple light-like Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory, focussing primarily on tree-level calculations and, beyond tree-level, to the Abelian theory. Here we calculate $O(g^2)$ correlators of multiple light-like Wilson loops in the $SU(N)$ theory. We use the chiral box expansion and a study of the leading singularities of the loop integrand to arrive at integrated expressions for these objects. We then use the results of these calculations to verify that a natural generalisation of the $\bar{Q}$-equation, familiar from the study of single Wilson loops, holds in the $SU(N)$ theory. This $\bar{Q}$-equation should provide a valuable tool for the computation of multiple Wilson loop correlators at higher order in the coupling.

💡 Research Summary

In this paper the authors investigate correlation functions of multiple light‑like Wilson loops in four‑dimensional 𝒩=4 supersymmetric Yang‑Mills theory, focusing on the first non‑trivial quantum correction, i.e. the order‑g² contribution, in the full SU(N) gauge group rather than the planar limit or the Abelian theory. The motivation stems from the well‑known duality between single light‑like Wilson loops and scattering amplitudes, and from the existence of a Ward‑type \bar{Q} equation that relates the action of a supersymmetry generator \bar{Q} on a Wilson loop expectation value to a correlator with an insertion of the chiral Lagrangian. While this equation has been extensively used in the planar (large‑N) setting for single loops, its validity for multiple loops and beyond the planar limit has not been established.

The authors first set up the problem in twistor space. They employ the twistor action for 𝒩=4 SYM, which separates into a holomorphic Chern‑Simons part S₁ (the self‑dual sector) and a non‑self‑dual part S₂ that generates insertions of a “Lagrangian line” X in twistor space. The supersymmetric Wilson loop operator is written as a trace of a path‑ordered exponential of the full super‑connection A along a polygonal contour whose edges are represented by intersecting CP¹ lines. The propagator for A in axial gauge is given explicitly, and the coupling constant is expressed in terms of the ’t Hooft coupling g² = g²_{YM} N/(16π²). With these ingredients the O(g²) contribution to a correlator of m Wilson loops is obtained by expanding S₂ to first order, which amounts to inserting a single Lagrangian line and contracting it with the Wilson loops via twistor propagators.

A set of diagrammatic rules is then derived. Only diagrams in which at least two propagators end on the Lagrangian line contribute, and each external twistor line must have at least one propagator attached. Diagrams with propagators running between adjacent twistor lines or with multiple propagators between the same pair of lines vanish due to Grassmann counting. The authors translate each diagram into an explicit integral over parameters s_{x,i} (positions on the Lagrangian line) and s_{i,k} (positions on the external lines), together with the standard twistor propagator factor. This yields compact expressions (3.4)–(3.6) for the integrand.

To perform the actual integration the authors adopt the chiral box expansion introduced in earlier work. At order g² any loop integral can be decomposed into a sum of chiral box (or triangle) integrals, each multiplied by a coefficient that is a leading singularity of the original integrand. The leading singularities are computed directly from the twistor diagrams and turn out to be simple rational functions of the external twistors, essentially tree‑level objects. By matching these coefficients with known results for single Wilson loops the authors verify that the same chiral box basis works for multiple‑loop correlators, with the only new feature being the appearance of mixed chiral pentagon‑type contributions when more than two loops are involved.

Having obtained explicit integrated expressions for the O(g²) correlators, the authors turn to the generalized \bar{Q} equation. The conjectured form relates the action of the supersymmetry generator \bar{Q} on an m‑loop correlator at Grassmann degree k to a sum over correlators with one additional Lagrangian insertion (i.e. degree k+1) and a specific rational kernel. The authors test the equation for the simplest non‑trivial case k=1. They compute the left‑hand side by acting with \bar{Q} on the O(g⁰) (tree‑level) correlator, and they compute the right‑hand side by integrating the O(g²) correlator with the kernel prescribed by the equation. Both sides are evaluated using the chiral box results and the explicit leading singularities. The two sides match exactly, confirming that the generalized \bar{Q} equation holds in the full SU(N) theory at order g², not only in the planar limit.

The paper concludes by emphasizing that the combination of the chiral box expansion and leading‑singularity analysis provides a practical method for evaluating multiple Wilson loop correlators beyond the planar approximation. Moreover, the existence of a non‑planar \bar{Q} equation suggests a recursive strategy for higher‑loop calculations: higher‑order corrections can be reduced to lower‑order ones via the supersymmetry Ward identity, just as has been successfully employed for single Wilson loops up to three loops. The authors indicate that a forthcoming companion paper will present a systematic classification of leading singularities at O(g²) for arbitrary numbers of loops, and they anticipate that the techniques developed here will enable future computations at O(g⁴) and beyond, opening the way to a deeper understanding of non‑planar dynamics in maximally supersymmetric gauge theory.