Multi-Loop Negative Geometries

Abstract: Scattering amplitudes in planar ${\cal N}=4$ supersymmetric Yang-Mills theory are dual to expectation values of null polygonal Wilson loops. The Amplituhedron provides a geometric construction for the all-loop integrand as the canonical form on the geometric region in the Grassmannian defined by a certain set of inequalities. For a closely related object, the logarithm of the scattering amplitude, the integrand is reproduced in a similar way using negative geometries. When integrated over all loop momenta except one, the result is infrared (IR) finite and equal to the expectation value of a certain Wilson loop with a Lagrangian insertion. At four points, this quantity, ${\cal F}(g,z)$ only depends on a single cross ratio $z$ and the 't Hooft coupling $g$. At weak coupling, it is known up to three loops from perturbative Wilson loop computations and at strong coupling through the AdS/CFT correspondence at leading order. In this paper, we explore this object further through the lens of the Amplituhedron and negative geometries, which provide very natural IR finite building blocks. We perform an explicit three-loop computation of all negative geometries and show that the number of internal cycles in the diagram is closely linked to the depth of polylogarithms. We calculate the cusp anomalous dimension $Γ_{\rm cusp}$ by integrating ${\cal F}(g,z)$ over $z$. We show that the higher-cycle diagrams are suppressed if we consider separate odd and even zeta contributions. Furthermore, we focus on certain convergent infinite series of one-cycle diagrams, perform all-loop order resummations of such contributions, and discuss various features of the result.

• The paper completes the canonical negative-geometry integrand basis through four loops, constructing the new three-cycle numerator from 28 building blocks fixed by double-pole, spurious-cut, and residue constraints.
• Multi-loop graph cycles control analytic complexity: tree geometries produce logarithms, while one-cycle and higher-cycle graphs generate deeper polylogarithms and multiple-zeta values that cancel in physical summed observables.
• The paper resums ladders, trees, triangle-ladders, box-ladders, and related classes, finding that low-cycle graphs dominate cusp-anomalous-dimension sectors while some one-cycle sums show nonphysical strong-coupling growth.

Overview and motivation

This paper develops the negative geometry expansion of planar $\mathcal{N}=4$ super Yang-Mills amplitudes, applying it to the four-point Wilson loop with a Lagrangian insertion (WLI), denoted ${\cal F}(g,z)$. This observable is the logarithm of the amplitude with one loop momentum frozen; it is infrared finite, has uniform maximal transcendental weight, depends on a single cross ratio $z$ and the 't Hooft coupling $g$, and is one integration away from the full amplitude. Because it is finite term by term in the negative-geometry basis, it can be computed entirely in $D=4$ without dimensional regularization — a property the authors exploit throughout.

The central objects are canonical dlog forms on "negative geometries": configuration spaces of $L$ lines $AB_i \in G(2,4)$ satisfying the one-loop Amplituhedron inequalities together with mutual conditions $\langle AB_i AB_j\rangle < 0$ for each edge of a graph $G$. Expanding each positive link of the Amplituhedron as (no link) minus (negative link) yields an expansion of the amplitude integrand as a signed sum over such geometries; taking connected graphs gives the integrand of the amplitude's logarithm. The graphs are organized by their number of internal cycles $\mathcal{C}$, with ${\cal F}(g,z)$0. The paper's two main threads are an explicit three-loop computation of all contributing negative geometries, and all-loop resummations of selected one-cycle classes.

Integrands at four loops

The numerator of any negative-geometry form is constrained by three types of data: absence of double poles under subsequent residues, vanishing when geometric inequalities are violated, and reduction to known forms on residues. Tree numerators factorize as products of the two-point building block ${\cal F}(g,z)$1 over all links; one-cycle cores require a compensating remainder ${\cal F}(g,z)$2 to restore logarithmicity, with branches again multiplying by ${\cal F}(g,z)$3 factors. The genuinely new result is the construction of the three-cycle integrand at ${\cal F}(g,z)$4: its numerator decomposes into the naive product, embedded one-cycle remainders (${\cal F}(g,z)$5: 4 terms; ${\cal F}(g,z)$6: 3 terms; ${\cal F}(g,z)$7: 6 terms), and a single irreducible piece ${\cal F}(g,z)$8, which is shown to vanish on every cut where any ${\cal F}(g,z)$9 has definite sign. The ansatz involves 28 building blocks in $z$0, $z$1, and mixed $z$2 sectors built from the $z$3 structures; solving double-pole and spurious-cut constraints fixes all coefficients uniquely (up to two relations among basis elements). With this, the canonical forms for all $z$4 negative geometries are complete.

The authors conjecture that these three constraint types always fix the numerator uniquely, but they concede that the sufficient list of constraints for a generic geometry is not known a priori, and that direct triangulation of the geometries remains out of reach with current technology.

Integrated geometries and polylogarithm depth

Freezing one loop (the marked point) and integrating the rest produces IR finite functions $z$5 of uniform weight $z$6. Two computational tools dominate:

• The boxing equation: for geometries where the marked point attaches through a single link, the Laplacian $z$7 removes that link, $z$8, with boundary condition $z$9. All tree graphs reduce to this equation, and tree integrands factorize into products of solvable pieces.
• Direct integration via canonical differential equations on a 257-element uniform-transcendentality basis (with IBP reduction via NeatIBP), checked numerically against AMFlow and pySecDec.

A structurally important finding is that the number of graph cycles controls the depth of the resulting polylogarithms. Tree graphs yield only powers of $g$0 (depth zero); the one-cycle triangle at weight four already contains the depth-two function $g$1, visible in its symbol through terms like $g$2.

At three loops, six negative-geometry integrands generate eleven integrated contributions. Individual graphs contain the spurious symbol letter $g$3, which first appears at the fourth symbol entry; after summing all contributions, $g$4 cancels and the total symbol agrees exactly with the Wilson-loop result of Henn et al. This agreement is strong evidence for the correctness of the negative-geometry expansion conjecture. The cancellation mechanism itself — how it is encoded in the cuts of the integrands — is left as an open question.

Resummations: ladders, trees, triangle-ladder, box-ladder

The ladder series satisfies $g$5 and resums to

$g$6

All tree graphs exponentiate into a generating functional obeying $g$7, solved by $g$8; notably, $g$9 at strong coupling, which does not reproduce the leading $D=4$0 behavior but preserves the $D=4$1 expansion.

For one-cycle classes, the paper resums the triangle-plus-ladder series $D=4$2, defined recursively by $D=4$3 with explicit closed-form solutions in HPLs whose constants $D=4$4 are fixed recursively by the $D=4$5 boundary condition. The full series obeys an inhomogeneous differential equation with source $D=4$6, solved perturbatively via Mellin transform. Analogous results hold for the box-plus-ladder class. A key qualitative distinction emerges at strong coupling: whereas ladders and trees behave linearly in $D=4$7 (matching the physical $D=4$8), the triangle-ladder sum grows as $D=4$9 and its cusp contribution as $L$0 — faster than the physical answer but slower than the weak-coupling leading term ($L$1). The authors call such series partially summable, and argue that adequate strong-coupling suppression likely requires embedding these classes into larger sets of diagrams starting at lower perturbative order. Whether the sum of all one-cycle geometries behaves as $L$2 at strong coupling, or must be combined with higher-cycle geometries, is explicitly left open.

Geometric-series constructions provide super-summable examples: the "sun" series sums to $L$3, exponentially suppressed except near $L$4 where the exponent vanishes, so its cusp contribution decays only logarithmically, $L$5. Gluing one-cycle triangles instead gives $L$6, whose cusp contribution is genuinely exponentially suppressed, $L$7.

Cusp anomalous dimension and cycle hierarchy

Using the contour-integral operator $L$8, the cusp anomalous dimension follows from $L$9 without any regularization. Applying shuffle identities reduces every contribution to convergent sums evaluable in Gamma functions. The per-graph results reveal a hierarchy: at three loops, tree graphs contribute $AB_i \in G(2,4)$0 against the one-cycle correction $AB_i \in G(2,4)$1 (full result $AB_i \in G(2,4)$2); at four loops, splitting into even-zeta ($AB_i \in G(2,4)$3) and odd sectors shows that lower-cycle graphs dominate within each sector separately — trees give $AB_i \in G(2,4)$4 of the $AB_i \in G(2,4)$5 coefficient, one-cycles $AB_i \in G(2,4)$6, two-cycles $AB_i \in G(2,4)$7, and the three-cycle only $AB_i \in G(2,4)$8, while in the $AB_i \in G(2,4)$9 sector trees vanish entirely and one-cycles contribute $\langle AB_i AB_j\rangle < 0$0 against $\langle AB_i AB_j\rangle < 0$1 (two-cycle) and $\langle AB_i AB_j\rangle < 0$2 (three-cycle).

Two structural observations carry significant implications. First, individual geometries produce multiple zeta values absent from the final answer: the triangle-plus-ladder class introduces $\langle AB_i AB_j\rangle < 0$3 at four loops, and gluing $\langle AB_i AB_j\rangle < 0$4 one-cycle triangles raises the maximal MZV depth by two per cycle (e.g., depth-four MZVs from the bow-tie square of $\langle AB_i AB_j\rangle < 0$5). Since $\langle AB_i AB_j\rangle < 0$6 contains only single zetas, a cancellation mechanism across geometries must exist; understanding it is proposed as a concrete handle on how integrability constrains amplitudes. Second, because $\langle AB_i AB_j\rangle < 0$7 is independent of the marked point's location, contributions can be assigned to unmarked graphs, enabling the infinite-class computations above.

Limitations and open problems

Several limitations are stated plainly. Integration technology is the principal bottleneck: beyond graphs solvable by the boxing equation, the authors rely on general canonical differential equations not tailored to negative geometries, making the four-loop computation of the full $\langle AB_i AB_j\rangle < 0$8 currently out of reach. No differential equation analogous to the Laplacian is known for geometries where the frozen node attaches through multiple links (e.g., the frozen corner of the three-loop triangle). The uniqueness conjecture for numerators lacks a proof, and the sufficient constraint set for generic geometries is unknown. The mechanism canceling both the spurious letter $\langle AB_i AB_j\rangle < 0$9 and higher-depth MZVs in the summed cusp is unidentified. Finally, no first-principles derivation of $G$0 from the BES equation via amplitudes exists; the connection remains conjectural despite extensive evidence.

Conclusion

The paper completes the negative-geometry integrand basis through four loops, establishes a correspondence between internal graph cycles and polylogarithm depth, verifies the three-loop WLI against Wilson-loop computations term by term, and delivers new all-loop resummations of one-cycle classes with controlled strong-coupling asymptotics. The demonstrated low-cycle dominance of $G$1 in separated zeta sectors, together with the mandatory cancellation of multiple zeta values, frames a concrete program: extend integration methods to generic negative geometries, determine whether fixed-cycle subclasses resum with physical strong-coupling behavior, and decode the cancellation mechanisms as a route toward manifest integrability in amplitudes.