BCFW recursion for light-like Wilson loop correlators

Abstract: We apply the idea of holomorphic linking (arXiv:1101.1329) to correlators of multiple light-like loop operators in $\mathcal{N}=4$ super Yang-Mills theory. This allows us to extend the BCFW recursion relations for planar scattering amplitude integrands or single Wilson loops in the planar theory to correlators of multiple loop operators. We discover a novel term which feeds the recursion for single loops into the relation for multiple loop operators. We illustrate the recursion relations with several examples and highlight its applicability in combination with the $\bar{Q}$-equation for multiple loop operators. Finally, we again employ holomorphic linking to derive a recursion relation for colour-exact Wilson loop correlators.

• The paper extends BCFW recursion for Wilson loops to correlators of two or more light-like Wilson loops in supersymmetric gauge theory.
• Single-loop recursion mechanism is derived via holomorphic linking in twistor space, linking the intersecting and factorising configurations to the variety of distance of the same region.
• All-loop integrand relations are obtained by including non-local log-det interactions, and closed recursion relations are derived for color-exact correlators beyond the planar limit.

Overview

This paper extends the BCFW recursion relations—originally derived for scattering amplitudes and, via the amplitude/Wilson loop duality, for single light-like Wilson loops in planar $\mathcal{N}=4$ super Yang-Mills—to correlators of multiple light-like Wilson loop operators. The mechanism is holomorphic linking in twistor space (Bullimore et al., 2011): Wilson loops are built from intersecting holomorphic lines in supertwistor space $\mathbb{CP}^{3|4}$, and deformations of these lines generate Makeenko-Migdal-type loop equations whose singular configurations (self-intersections) reproduce the factorisation channels of ordinary BCFW recursion. The paper derives tree-level recursions for connected correlators of two loops, promotes them to all-loop integrand relations by including the non-local "log-det" interactions completing the self-dual theory to full $\mathcal{N}=4$ SYM, and finally obtains a closed recursion for colour-exact Wilson loop correlators beyond the planar limit.

Twistor setup

The perturbative formulation uses a $(0,1)$-form superconnection $\mathcal{A}$ on $\mathbb{CP}^{3|4}$ with action $S = S_1 + S_2$, where $S_1$ is holomorphic Chern-Simons (the self-dual sector) and $S_2$ is the non-local log-det interaction on lines $X \subset \mathbb{CP}^{3|4}$:

$\mathbb{CP}^{3|4}$0

The couplings obey $\mathbb{CP}^{3|4}$1, with the specific choice $\mathbb{CP}^{3|4}$2 fixed by matching planar Wilson loop expectation values to amplitudes and ensuring the colour-exact Wilson loop obeys the $\mathbb{CP}^{3|4}$3-equation. Light-like loop operators are traces of holonomies around sequences of intersecting lines $\mathbb{CP}^{3|4}$4, with parallel transporters $\mathbb{CP}^{3|4}$5 obtained from the holomorphic frame solving $\mathbb{CP}^{3|4}$6.

Holomorphic linking and the single-loop recursion

For a one-parameter holomorphic family of curves $\mathbb{CP}^{3|4}$7, the $\mathbb{CP}^{3|4}$8-variation of the parallel transporter reduces to the pullback of the $\mathbb{CP}^{3|4}$9-form curvature $\mathcal{N}=4$0 onto the ruled surface swept out by the deformed line. Writing $\mathcal{N}=4$1 as the variational derivative of the Chern-Simons action and integrating by parts inside the path integral yields the loop equation

$\mathcal{N}=4$2

where $\mathcal{N}=4$3 ($\mathcal{N}=4$4) for U($\mathcal{N}=4$5) (SU($\mathcal{N}=4$6)). The delta function forces the two lines to intersect, splitting the contour into two sub-loops $\mathcal{N}=4$7, $\mathcal{N}=4$8. In the large-$\mathcal{N}=4$9 limit this factorises into products of lower-multiplicity expectation values; choosing the shift $(0,1)$0 and integrating against $(0,1)$1 gives the standard BCFW form

$(0,1)$2

with shifted twistors defined by line intersections, reproducing the result of Bullimore–Skinner.

Without taking the large-$(0,1)$3 limit, the relation is not closed: it involves the connected part of a two-loop correlator at a special kinematic configuration where both loops share an intersection twistor $(0,1)$4. This is precisely the object the paper then tackles directly.

Recursion for two-loop correlators

Varying the correlator of two loop operators produces three classes of terms: self-intersections within each loop (as before), intersections between the two loops, and SU($(0,1)$5)-specific subtraction terms. After decomposing into disconnected and connected pieces and using the single-loop equation to cancel the disconnected contributions, the large-$(0,1)$6 connected correlator satisfies

$(0,1)$7

The fourth term is novel: it feeds the single-loop recursion into the multi-loop relation through a Wilson loop in a self-intersecting configuration that traverses the first loop, passes through the intersection point $(0,1)$8, runs around the second loop, and returns. Notably, unlike the planar single-loop case, this recursion explicitly distinguishes U($(0,1)$9) from SU($\mathcal{A}$0).

As an application, the authors use these relations to justify the general analysis of the $\mathcal{A}$1-equation for multiple Wilson loop correlators (Drummond et al., 14 Dec 2025, Drummond et al., 30 Jan 2026), verifying the colour-exact $\mathcal{A}$2-equation at $\mathcal{A}$3: under the BCFW shift, only R-invariant factors carry the $\mathcal{A}$4 poles required by the collinear integral, while the correlator parts are smooth as $\mathcal{A}$5.

Loop-level integrands and the forward-limit term

Including $\mathcal{A}$6, the curvature insertion acquires an additional contribution from $\mathcal{A}$7, expressible via the current $\mathcal{A}$8. Rewriting $\mathcal{A}$9 as a double contour integral converts the variation into an insertion of a new loop operator on an auxiliary line $\mathbb{CP}^{3|4}$0, and after variable redefinitions the resulting term takes the canonical form of an R-invariant integrated over the GL(2) moduli of the auxiliary line:

$\mathbb{CP}^{3|4}$1

This matches the forward-limit term of the all-loop integrand recursion of Arkani-Hamed et al. (Arkani-Hamed et al., 2010). For two-loop correlators, the analogous term carries Lagrangian insertions distributed across partitions between the two loops. The GL(2) integral can be evaluated by residues: at one loop it yields Kermit diagrams, and at higher loops a sequence of BCFW expansions produces bridge factors multiplying products of lower-loop Wilson loops (Bourjaily et al., 2013, Bourjaily et al., 2023).

Consistency checks

Tree level. The recursion correctly predicts vanishing MHV connected correlators (iteration terminates on a backtracking loop). At NMHV it reproduces the known U($\mathbb{CP}^{3|4}$2) sum $\mathbb{CP}^{3|4}$3 and the vanishing SU($\mathbb{CP}^{3|4}$4) result. The N²MHV case is the most substantive check: in the SU($\mathbb{CP}^{3|4}$5) theory the connected tree-level N²MHV correlator is known to factorise as half the square of the U($\mathbb{CP}^{3|4}$6) NMHV expression—a fact far from manifest in the BCFW representation. The paper provides an analytic proof of this equivalence, first for triangles against $\mathbb{CP}^{3|4}$7-gons and then for general $\mathbb{CP}^{3|4}$8, using $\mathbb{CP}^{3|4}$9-supersymmetric gauge fixing and R-invariant identities. A delicate point here is the treatment of the self-intersecting term: naively divergent or ill-defined R-invariants (with repeated twistors or degenerate denominators on Plücker support) are regulated by perturbing the repeated twistor supersymmetrically; apparent divergences carry vanishing Grassmann factors provided the accompanying R-invariant is multiplied in before taking the limit—the same phenomenon as the forward-limit term in the single-loop all-loop recursion. Numerically, the recursion was verified at NMHV through N⁴MHV for two-loop correlators up to octagon-octagon, for both gauge groups.

Loop level. At $S = S_1 + S_2$0 MHV, the recursion's forward-limit term reduces to the product of two R-invariants integrating to Kermit diagrams $S = S_1 + S_2$1, matching the diagrammatic expansion of (Drummond et al., 14 Dec 2025) for all multiplicities. Numerical checks confirm the relation at NMHV up to octagon-octagon and N²MHV up to hexagon-square at $S = S_1 + S_2$2, and at MHV/NMHV up to pentagon-square at $S = S_1 + S_2$3.

Recursion beyond the planar limit

For colour-exact correlators of $S = S_1 + S_2$4 loops, the holomorphic-linking equation has a structural feature: an $S = S_1 + S_2$5-loop correlator depends on an $S = S_1 + S_2$6-loop correlator, so no single equation closes. However, since the extra correlator always enters at lower MHV degree (dressed by an R-invariant), iterating across MHV degree terminates. The resulting colour-exact recursion expresses any correlator in terms of lower-cusp-number or lower-Grassmann-degree objects. As an explicit example, the colour-exact tree-level N²MHV hexagon in U($S = S_1 + S_2$7) is computed recursively and found to relate to its planar part by

$S = S_1 + S_2$8

a simple cross-ratio-dependent correction verified numerically. This makes explicit a connection between non-planar corrections to a single Wilson loop and correlators of multiple Wilson loops at lower MHV degree—for instance, the colour-exact single-loop expectation value is built from two-loop correlators one MHV degree down.

Limitations and open questions

Several caveats attend these results. The derivation is formal: it relies on manipulations of the path integral (integration by parts of functional derivatives) without rigorous justification, and the regularisation prescription for self-intersecting configurations, while argued to be well-defined via supersymmetric limits, is a prescription rather than a derived property. The claim that the GL(2) forward-limit integral can be evaluated explicitly at arbitrary loop order rests on extrapolating the one- and higher-loop patterns of (Bourjaily et al., 2013, Bourjaily et al., 2023) rather than a proof. Open questions stated by the authors include: the precise growth rate of the number of terms in the BCFW versus twistor-diagram representations, and whether truncating the recursion at factorising cases (N²MHV tree, NMHV one-loop, MHV two-loop in SU($S = S_1 + S_2$9)) is advantageous; whether these recursions triangulate some generalisation of the amplituhedron to multiple loop operators; whether the BCFW representations admit cluster-adjacency structure for their poles, as holds for a single Wilson loop (Drummond et al., 2018); and whether planar leading-singularity formulae admit non-planar generalisations in terms of correlators rather than products of loop operators.

Conclusion

The paper demonstrates that holomorphic linking in twistor space yields BCFW-type recursion relations for connected correlators of multiple light-like Wilson loops, featuring a new "intersecting" term that couples the single-loop recursion to multi-loop correlators, extends to all-loop integrands via the log-det interactions, and terminates for colour-exact correlators through an interplay between loop number and MHV degree. Verified analytically—including the non-trivial perfect-square factorisation at N²MHV—and extensively numerically, these relations have already streamlined verification of the multi-loop $S_1$0-equation and provide a systematic computational route to non-planar Wilson loop data.