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

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 $O(g<sup>2)$ 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.

• Heavy copper
• The paper compute correlators of multiple light-like (super) Wilson loops in $SU(N) \mathcal{N}=4$ super Yang-Mills theory at order $O(g^{2})$
• The authors verify a conjectured generalization of the $ar{Q}$-equation to multiple Wilson loops at large $N$, with explicit numerical checks up to eight-sided polygons.
• In complex kinematics, integrands for correlators of triangular loops develop uncanceled $\log(\epsilon)$ divergences at N$^2$MHV.

Overview

This paper by Drummond, Rochford and Wright computes correlators of multiple light-like (super) Wilson loops in $SU(N)$ $\mathcal{N}=4$ super Yang-Mills theory at order $O(g^2)$, where $g^2 = g_{\rm YM}^2 N/(16\pi^2)$. Building on earlier tree-level work in the Abelian and planar theories, the authors derive integrated expressions for the connected parts of these correlators using the chiral box expansion of Bourjaily, Caron-Huot and Trnka, and use the results to verify a conjectured generalisation of the $\bar{Q}$-equation to multiple Wilson loops. The verification is carried out both for single Wilson loops beyond the planar limit (colour-exact) and for two-loop correlators at large $N$, with explicit numerical checks up to eight-sided polygons and up to N$^3$MHV.

The computational framework is the twistor super Wilson loop formalism of Mason and Skinner, with the perturbative expansion organised around the twistor action split into a holomorphic Chern-Simons (self-dual) piece $S_1$ and an interaction term $S_2$ given by a log-det over lines in supertwistor space. A notable technical refinement is the choice of scaling $\alpha = C_F/(2\pi^2)$ relating $\mathcal{N}=4$0 and $\mathcal{N}=4$1, which reduces to the planar choice at large $\mathcal{N}=4$2 but is required for the ordinary $\mathcal{N}=4$3-equation to hold for colour-exact Wilson loops.

Loop integrands at $\mathcal{N}=4$4

At $\mathcal{N}=4$5, computing the loop integrand amounts to evaluating the tree-level correlator of the Wilson loops with a single Lagrangian line $\mathcal{N}=4$6; higher orders would require more Lagrangian insertions. The authors give compact Feynman rules for writing down any such diagram: propagator insertions on the Lagrangian line carry cyclic denominators $\mathcal{N}=4$7, external twistor line insertions carry ordered denominators, and each propagator contributes a $\mathcal{N}=4$8 supported on five points including the reference twistor $\mathcal{N}=4$9. Each N$O(g^2)$0MHV diagram evaluates to a product of $O(g^2)$1 $O(g^2)$2-invariants times a rational prefactor. Diagram generation, colour factors, and evaluation are automated.

Two structural facts constrain the results. First, tracelessness of $O(g^2)$3 generators implies that the connected $O(g^2)$4 contribution vanishes below N$O(g^2)$5MHV for $O(g^2)$6 loops; for two loops the first non-trivial case is N$O(g^2)$7MHV, with four propagators. Second, the leading connected contributions carry a $O(g^2)$8 suppression relative to the disconnected product, as expected from the double-trace structure.

A striking kinematic result concerns triangular loops: in real kinematics, correlators involving triangles vanish diagram-by-diagram at all MHV degrees, even at integrand level — this extends the known MHV/NMHV vanishing to all degrees. In complex kinematics, however, the integrands are generically non-zero, and the collinear limit of an integrated square-square remainder to a triangle-square correlator develops uncanceled $O(g^2)$9 divergences at N$g^2 = g_{\rm YM}^2 N/(16\pi^2)$0MHV. The paper leaves open how such complex-kinematic triangle correlators should be regularised; the Schubert problems involved degenerate (a continuous one-parameter family of solutions replaces the usual discrete pair), so the chiral box machinery does not directly apply.

Local integrands via the chiral box expansion

The conversion from diagrammatic integrands to integrated answers proceeds through the chiral box basis. For each physical quadruple cut there are two Schubert solutions for the loop line $g^2 = g_{\rm YM}^2 N/(16\pi^2)$1, and the two conjugate chiral box integrands have residues $g^2 = g_{\rm YM}^2 N/(16\pi^2)$2 on one solution and zero on the other. The coefficients are therefore the leading singularities of the integrand on each Schubert solution, computed explicitly here by multidimensional residue extraction from the sum of diagrams. Because the chiral boxes are finite while the full answer is UV divergent, divergent triangle integrals must be added; their coefficient is fixed both by the divergence structure and by the requirement of cancelling spurious two-mass-hard cuts involving the auxiliary bitwistor $g^2 = g_{\rm YM}^2 N/(16\pi^2)$3. That coefficient is simply the N$g^2 = g_{\rm YM}^2 N/(16\pi^2)$4MHV tree-level connected correlator. In the remainder function, obtained by subtracting the tree-level connected part times the summed one-loop MHV contributions of each individual loop, the triangles cancel entirely.

An important bookkeeping subtlety arises for multiple loops: there is no canonical ordering of poles in a cut mixing propagators from different loops, and permuting pole order flips the sign of the residue. The authors resolve this by labelling boxes by their propagators rather than legs and fixing a pole ordering convention. They also extend the chiral box basis to include zero-mass boxes, which are irrelevant for single Wilson loops but essential when squares appear.

As a check of consistency, the MHV sector reproduces the familiar chiral pentagon expansion, with each pentagon identified as a specific chirality of a two-mass-easy, one-mass or zero-mass chiral box depending on leg separation, and degenerating to minus the triangle integral for adjacent legs.

The paper presents explicit results in ancillary files for the N$g^2 = g_{\rm YM}^2 N/(16\pi^2)$5MHV square-square, square-pentagon and pentagon-pentagon remainders and the N$g^2 = g_{\rm YM}^2 N/(16\pi^2)$6MHV square-square remainder, with box coefficients expressed in terms of $g^2 = g_{\rm YM}^2 N/(16\pi^2)$7-invariants and NMHV tree-level Wilson loops evaluated on degenerate configurations (interpreted via limits of decagons, with divergent terms eliminated for Grassmann reasons). Auxiliary-twistor independence of the final expressions holds only after imposing 33 linear relations among the 70 leading singularities of the square-square case — it is not manifest term by term.

The $g^2 = g_{\rm YM}^2 N/(16\pi^2)$8-equation beyond the planar limit

Before addressing multiple loops, the paper verifies that the original single-loop $g^2 = g_{\rm YM}^2 N/(16\pi^2)$9-equation holds for colour-exact Wilson loops, not merely planar amplitudes. At $\bar{Q}$0 and $\bar{Q}$1 this follows from a decomposition of the non-planar tree-level N$\bar{Q}$2MHV contribution into planar and Abelian pieces, combined with the identity $\bar{Q}$3, which itself rests on pairwise cancellation between planar and reflected non-planar three-insertion diagrams. For general $\bar{Q}$4, the right-hand side is integrated using a colour-exact BCFW-type recursion relation adapted from Bullimore-Skinner, isolating all $\bar{Q}$5 poles into single $\bar{Q}$6-invariant prefactors whose collinear integrals are known. A numerical check succeeds for $\bar{Q}$7, $\bar{Q}$8. This establishes that the equation is sensitive to the normalisation of the twistor action but holds as written under the chosen $\bar{Q}$9 scaling.

Verification for two Wilson loops

For two loops at large $N$0, the relevant relation connects $N$1 acting on the connected $N$2 N$N$3MHV remainder to collinear integrals of connected tree-level N$N$4MHV correlators, with insertion of an extra vertex on either loop. On the left-hand side, $N$5 annihilates the box coefficients and acts only on the dilogarithms and logarithms of the integrated chiral boxes, producing terms of the form $N$6. On the right-hand side, a BCFW shift $N$7 expands the $N$8-loop correlator into products of lower-degree correlators dressed by single $N$9-invariants; only $^3$0-invariants of four specific forms survive the collinear integration, and apparent $^3$1 poles cancel non-trivially between terms (equivalently, between two different BCFW shifts).

Both sides reduce to linear combinations of $^3$2 structures, and the authors verify equality numerically for all cases with N$^3$3MHV or N$^3$4MHV left-hand sides and multiplicities up to eight sides per loop. This constitutes the main evidence that the conjectured multi-loop $^3$5-equation holds in the interacting $^3$6 theory, extending the previous checks which covered only the Abelian theory and factorised large-$^3$7 pieces. The equation is expected to hold non-perturbatively, but only the $^3$8 level has been tested here.

Limitations and open questions

Several limitations are stated plainly. The explicit $^3$9 computations are performed in the planar limit for the multi-loop checks; the colour-exact verification covers only single Wilson loops. Triangle correlators at N$S_1$0MHV in complex kinematics are genuinely divergent and lack a regularisation scheme within the current framework. Compact formulae for the leading singularities are deferred to a companion paper, as is the derivation of the BCFW recursion relations used on the right-hand side of the $S_1$1-equation. Beyond $S_1$2, local integral bases exist but their integrated forms are generally unknown, so extending the program to $S_1$3 and $S_1$4 relies entirely on bootstrapping through the $S_1$5-equation; whether elliptic integrals appear along this route is unresolved. Finally, the analytic structure of these correlators — symbol alphabets, cluster adjacency, and the appropriate modification of the positive Grassmannian given the modified cyclic symmetry — remains unexplored.

Conclusion

This work renders the $S_1$6 problem for multiple light-like Wilson loop correlators fully tractable via the chiral box expansion, provides explicit integrated remainders for representative two-loop cases, and supplies substantial evidence — including non-planar single-loop checks and numerical two-loop verifications through N$S_1$7MHV — that the generalised $S_1$8-equation holds in the $S_1$9 theory. The equation now stands as the principal tool for pushing these correlators to higher loop order, mirroring its role for single Wilson loops, and may also bear on the structure of Wilson-loop/Lagrangian-insertion correlators, where restricted final entries suggestive of a descent equation have been observed at two loops.