Leading singularities of Wilson loop correlators from twistor Wilson loop diagrams

Abstract: The leading singularities of one-loop scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory are known to factorise into products of tree-level amplitudes, and this can be seen from a number of different perspectives e.g. generalised unitarity or on-shell diagrams. Here we investigate the leading singularities from the perspective of the Wilson loop expectation values to which these amplitudes are dual, in particular making use of the twistor Wilson loop formalism. We show that the factorisation of one-loop leading singularities of a null Wilson loop's expectation value into a product of tree-level objects is manifest at the level of twistor Wilson loop diagrams, and is a simple consequence of planarity, without appeal to e.g. unitarity on the amplitude side of the duality. We then use the same approach to derive compact formulae for the one-loop leading singularities of correlators of multiple light-like Wilson loop operators in terms of tree-level objects. Via the chiral box expansion, these formulae provide a simple route to writing down the $O(g^2)$ correlation function of any number of Wilson loops at any MHV degree.

• The paper demonstrates that one-loop leading singularities factorize into products of tree-level correlators using the twistor Wilson loop formalism.
• It provides explicit diagrammatic and algebraic formulations for various mass box cuts and extends the analysis to multi-loop configurations.
• The work establishes a foundational framework linking Wilson loop correlators and amplitude unitarity through geometric twistor space methods.

Leading Singularities of Wilson Loop Correlators via Twistor Wilson Loop Diagrams

Overview

This work provides a comprehensive analysis of leading singularities in one-loop correlators of light-like Wilson loop operators in planar $\mathcal{N}=4$ SYM, employing the twistor Wilson loop formalism. The principal result is a manifestly planar, diagrammatic derivation demonstrating the factorization of one-loop leading singularities of null Wilson loop expectation values—traditionally interpreted via scattered amplitude unitarity—into tree-level objects at the level of twistor diagrams. This approach is further extended to multi-loop correlators, and compact, explicit formulae for arbitrary loop number and MHV degree are derived. The results enable immediate construction of the $\mathcal{O}(g^2)$ sector of light-like Wilson loop correlators for any multiplicity within the chiral box expansion framework.

Twistor Wilson Loop Formalism and Tree-Level Structure

The analysis is performed using the twistor Wilson loop formalism in twistor space, parameterizing light-like polygonal contours via sequences of projective lines and associated supertwistors. Within this framework:

• At tree level, connected correlators of $m$ Wilson loops in the $\text{N}^k$MHV sector are computed by planar twistor diagrams with $k$ propagators, each with Grassmann degree 4.
• Planar diagrams correspond to non-crossing propagator configurations in the 'outside' representation, with additional constraints for multiple Wilson loops: each loop must host at least two propagators to separate loops due to the vanishing color trace otherwise.
• The combinatorial rules for constructing the planar twistor diagrams are explicitly provided; they generalize previous results for amplitudes and single-loop expectations to fully connected multi-loop observables.

This structure is critical in organizing both tree-level and cut diagrams and sets the stage for the subsequent analysis of loop-level leading singularities.

One-Loop Structure: Lagrangian Insertions and Leading Singularities

At $\mathcal{O}(g^2)$, the relevant objects are diagrams with propagators to an auxiliary "Lagrangian line," representing the insertion of the field strength at a spacetime point, with a minimum of two such connections to avoid redundant color factors. The one-loop integrand is given in terms of such diagrams and can be systematically related to combinations of $R$-invariants, paralleling the known amplitude side story.

The leading singularities arise as residues on maximal cuts of the loop momentum, realized as conditions on four-brackets $\langle AB\, p\, q \rangle = 0$ for the Lagrangian line $(AB)$ intersecting four specified twistor lines. Classical Schubert calculus guarantees two such solutions ("Schubert solutions") for generic configurations, mirroring the two-fold ambiguity in leading singularities for maximally-cut boxes.

Manifest Factorization and All-MHV Degree Results

A central result is that, in the planar limit and for four-mass-type box singularities, the leading singularity naturally factorizes into the product of four tree-level correlators (or amplitudes), each associated with a segment ("region") partitioned by the cut propagators. This is proven at the level of twistor diagrams without recourse to unitarity or amplitude side duality. The argument is constructed as follows:

• For a single Wilson loop, the four-mass box cut leaves four regions, each supporting further tree-level structure; the total leading singularity is given by the product of corresponding tree-level expressions, dressed by a specific bosonic prefactor depending on the Schubert solution.
• At higher MHV degree, additional propagators are to be assigned within regions, with planarity enforcing a block-diagonal structure—no crossing between regions.
• The formulae for lower-mass cuts (e.g., two-mass easy/hard, one-mass, zero-mass) emerge as smooth collinear limits from the four-mass case, maintaining the factorization structure but varying the number and nature of tree-level factors.

Extension to Multi-Loop Correlators

For correlators of multiple Wilson loops, the structure generalizes. The classification of singularities is exhaustive: the distribution of the four cut propagators over the constituent loops induces types such as 4-0, 3-1, 2-2, 2-1-1, and 1-1-1-1 splits, each with precise combinatorial rules:

• The seed diagrams for a given split are explicitly listed and classified.
• For each configuration, the leading singularity is expressed as a sum over products of tree-level correlators and bosonic prefactors; these depend on the particular partition, the Schubert solution, and the color trace structure.
• The effect of spectator Wilson loops is handled via ordered partitions, replacing factors in the product over regions with connected parts of correlators involving those spectators.

Explicit formulae are presented for all cases, including the generalization to arbitrary numbers of spectator loops and regions.

Explicit Results and Formal Structure

The paper systematically organizes the results as follows:

• Closed-form expressions for all four-mass leading singularities in terms of intersection twistors associated with the Schubert solutions, cross-ratios, and Gram determinant-type square roots.
• Detailed recursive and partition-based formulae for all splits and configurations, including rules for handling limits, spectator loops, and vanishing-trace diagrams.
• Full appendices listing lower-mass leading singularities, establishing their derivation both via direct twistor diagrammatics and via degenerate limits of the general four-mass case.

Numerical coefficients and examples are provided throughout, and comparisons are made to known chiral box expansion results in the amplitude literature, confirming the formalism coherently reproduces all known planar one-loop correlator data.

Implications and Outlook

This formalization establishes a strict diagrammatic dual to the generalized unitarity picture used for amplitudes, but fully transferred to the Wilson loop side, confirming that factorization "comes from planarity" at the level of twistor diagrams without requiring explicit quadratic unitarity cuts. The practical implications are immediate: one can write down the $\mathcal{O}(g^2)$ connected part of any multi-loop correlator at any MHV degree explicitly.

This development paves the way for several directions:

• Systematic investigation of higher-loop leading singularities in the twistor diagrammatic language, possibly using the constructed framework as a starting point for multi-loop recursion.
• Potential formulation of non-planar leading singularities via extensions of the partition logic; though planarity is necessary for exact factorization, the underlying connection between maximal-cut diagrams and tree-level data suggests non-planar generalizations may be expressible as sums over connected trees with modified color combinatorics.
• Further exploration of Grassmannian integral formulations for multi-loop correlators, analogous to the single-loop, on-shell diagram approach, leveraging the explicit geometric structure uncovered.

The technical tools and explicit formulae derived herein now constitute a foundational reference for further development of analytic techniques in maximally supersymmetric gauge theory, particularly for observables beyond the amplitude sector.

Conclusion

The paper demonstrates that leading singularities of null Wilson loop correlators in planar $\mathcal{N}=4$ SYM are, at one-loop, fully determined by planar partitioning and twistor diagrammatics—realizing factorization into tree-level objects before recourse to amplitude unitarity or duality. The approach provides all necessary ingredients to generate, via chiral box expansions, the $\mathcal{O}(g^2)$ data for arbitrary multi-Wilson loop correlators at any MHV degree. The arguments are robustly extended to all possible configurations, with explicit formulae and a clear algebraic structure for handling arbitrary combinations of loops and external insertions. This work establishes the twistor Wilson loop formalism as a complete, powerful method for deriving and organizing all one-loop leading singularities of Wilson loop correlators in planar theory.