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

James Drummond, Matthew Rochford, Rowan Wright

TL;DR

This work advances the study of correlators of multiple light-like Wilson loops in $\mathcal{N}=4$ SYM by computing $O(g^2)$ contributions in the $SU(N)$ theory and expressing them through a chiral box expansion anchored in leading singularities. It demonstrates that a natural generalization of the $\bar{Q}$-equation holds beyond the planar limit, connecting integrated loop correlators to tree-level data and enabling higher-order computations. The analysis leverages twistor-space formalisms, a supersymmetric Wilson loop construction, and a detailed treatment of divergent triangle contributions to yield finite remainders and explicit box coefficients for complex multi-loop configurations. The results provide a practical framework for exploring the nonplanar structure of Wilson loop correlators and suggest broader applicability of the $\bar{Q}$-equation to other observables in four-dimensional $\mathcal{N}=4$ SYM.

Abstract

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.

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

TL;DR

This work advances the study of correlators of multiple light-like Wilson loops in SYM by computing contributions in the theory and expressing them through a chiral box expansion anchored in leading singularities. It demonstrates that a natural generalization of the -equation holds beyond the planar limit, connecting integrated loop correlators to tree-level data and enabling higher-order computations. The analysis leverages twistor-space formalisms, a supersymmetric Wilson loop construction, and a detailed treatment of divergent triangle contributions to yield finite remainders and explicit box coefficients for complex multi-loop configurations. The results provide a practical framework for exploring the nonplanar structure of Wilson loop correlators and suggest broader applicability of the -equation to other observables in four-dimensional SYM.

Abstract

In recent work we began a study of the correlators of multiple light-like Wilson loops in super Yang-Mills theory, focussing primarily on tree-level calculations and, beyond tree-level, to the Abelian theory. Here we calculate correlators of multiple light-like Wilson loops in the 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 -equation, familiar from the study of single Wilson loops, holds in the theory. This -equation should provide a valuable tool for the computation of multiple Wilson loop correlators at higher order in the coupling.
Paper Structure (26 sections, 157 equations, 3 figures)

This paper contains 26 sections, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup and conventions
  3. Multiple light-like Wilson loop correlators
  4. Super Wilson loops
  5. $\mathcal{N}=4$ super Yang-Mills theory in twistor space
  6. The supersymmetric Wilson loop
  7. Integrands at $O(g^2)$
  8. Loop integrands from twistor diagrams
  9. Finite remainder functions
  10. Comment on integrands involving triangular Wilson loops
  11. Local expressions for the integrand and integrated correlator
  12. A review of the chiral box expansion
  13. Triangle integrands
  14. MHV contributions at $O(g^2)$ and equivalence with the chiral pentagon expansion
  15. Chiral boxes for multiple Wilson loop correlators
  16. ...and 11 more sections

Figures (3)

  • Figure 1: A twistor diagram which contributes at N${}^2$MHV to a pentagon-square correlator at $O(g^2)$. Note that this diagram is planar, as can be seen heuristically by the fact that, drawing the Lagrangian line beneath the two Wilson loops and drawing all propagators outside the Wilson loops, it is possible to draw all of the propagators outside the Wilson loops without any crossing.
  • Figure 2: The standard notation for the three-mass quadruple cut corresponding to $\langle AB12 \rangle = \langle AB23 \rangle = \langle AB56 \rangle = \langle AB78 \rangle = 0$. For a single Wilson loop there is a natural ordering in which to put the legs, corresponding to the cyclic ordering of the Wilson loop itself.
  • Figure 3: Two natural ways of drawing the same three mass cut as in Fig. \ref{['octagonBox']}, but for a square-square correlator rather than a single octagon. There is no longer a natural ordering for the propagators being cut, nor is there a natural identity for the legs between labels on different Wilson loops.