Ladder Negative Geometries in SYM and ABJM

• Ladder negative geometries are chain-connected configurations arising from replacing mutual positivity with negativity in amplituhedron decompositions.
• In planar N=4 SYM and ABJM theories, they enable precise IR-finite results and all-loop recursion through tractable canonical forms and Mellin-space methods.
• They provide a benchmark for geometric analyses by linking maximal cuts, chamber decompositions, and leading singularity classifications to integrand constructions.

Searching arXiv for papers on ladder negative geometries and closely related negative-geometry/amplituhedron work. Ladder negative geometries are chain-like connected negative geometries that arise in amplituhedron-based decompositions of loop integrands for the logarithm of scattering amplitudes and for related Wilson-loop observables with a Lagrangian insertion. In planar $\mathcal N=4$ super-Yang–Mills theory, they are linear-chain configurations of loop variables subject to mutual negativity along adjacent links; in ABJM theory, they appear as ladder-type bipartite chains in the negative-geometry decomposition of the four-point amplitude logarithm. Across these settings, they are important because their canonical forms are explicitly tractable, their integrated representatives are infrared finite after integrating all but one loop variable, and they admit all-loop treatments based on recursion, differential equations, Mellin transforms, geometric Landau analysis, and symbol bootstrap (2112.06956, Li, 2024, Glew et al., 2024, Chicherin et al., 7 Aug 2025).

1. Definition and graph-theoretic formulation

Negative geometries are obtained by replacing some mutual positivity conditions of amplituhedron-type constructions with mutual negativity conditions. In the four-point formulation of planar $\mathcal N=4$ super-Yang–Mills, the usual amplituhedron imposes mutual positivity between loop lines,

$(AB)_i(AB)_j>0,$

whereas a negative geometry imposes

$(AB)_i(AB)_j<0$

for the edges of a chosen connected graph $G$. The connected negative geometries give the integrand for $\log M$, and this organization is geometric rather than a Feynman-diagram expansion (2112.06956).

In the loop-space formulation for MHV$_n$ amplitudes, each loop momentum $y_a$ is represented as a circular vertex, a green edge means positive separation, and a red edge means negative separation. The full $L$-loop amplituhedron is the complete graph on $L$ loop vertices with all green edges, while a ladder geometry is a chain of vertices with edges all chosen positive or all chosen negative. The external data are a null polygon $\mathcal N=4$0 in dual-momentum space $\mathcal N=4$1, and each loop variable lies in the one-loop fiber

$\mathcal N=4$2

A negative ladder is then the chain

$\mathcal N=4$3

with the positive-ladder analogue obtained by reversing the sign choice (Glew et al., 2024).

In ABJM theory, the decomposition of the four-point all-loop integrand into negative geometries retains only connected bipartite graphs,

$\mathcal N=4$4

Within this decomposition, a ladder-type negative geometry is the special subclass where the graph is a chain or ladder of alternating black and white nodes. This is the simplest all-loop family treated systematically in that setting (Li, 2024).

2. Canonical forms and all-loop factorization in loop space

The most explicit all-loop description of positive and negative ladders in loop space is given for planar $\mathcal N=4$5 SYM MHV$\mathcal N=4$6 amplitudes. A central ingredient is the chamber decomposition of the one-loop fiber $\mathcal N=4$7. Its vertices are

$\mathcal N=4$8

where the quadruple-cut points satisfy

$\mathcal N=4$9

Each chamber $(AB)_i(AB)_j>0,$0 is characterized by a fixed sign pattern of $(AB)_i(AB)_j>0,$1, and the one-loop canonical form decomposes as

$(AB)_i(AB)_j>0,$2

This chamber structure is then iterated along the ladder (Glew et al., 2024).

For the all-negative ladder on $(AB)_i(AB)_j>0,$3 loop vertices $(AB)_i(AB)_j>0,$4, the canonical form is written as a sum over quadruple cuts $(AB)_i(AB)_j>0,$5, with a left factor, a recursively defined middle factor, and a right factor. The same structure applies to positive ladders after replacing every red link by green and replacing $(AB)_i(AB)_j>0,$6 by $(AB)_i(AB)_j>0,$7. The paper emphasizes that the resulting formula expresses ladder contributions as sums over maximal cuts, with each term factorizing into products of either chiral pentagons or their simple generalizations (Glew et al., 2024).

The factorization extends the previously known two-loop “fibration of fibration” structure to arbitrary loop order. The basic one-loop and two-loop ingredients already exhibit this pattern: chamber forms multiply positive or negative fibers, and the higher-loop ladder recursively glues together these one-loop building blocks. For $(AB)_i(AB)_j>0,$8, the one-loop fiber has only one chamber and the formulas reduce to the original four-point negative-ladder expressions; for $(AB)_i(AB)_j>0,$9, there are $(AB)_i(AB)_j<0$0 chambers, reproducing the previously studied five-point negative ladders. The paper explicitly characterizes the all-loop ladder formula as a higher-loop analogue of the chiral pentagon expansion of the one- and two-loop momentum amplituhedron (Glew et al., 2024).

3. Four-point ladders in planar $(AB)_i(AB)_j<0$1 super-Yang–Mills

At four points, negative geometries reorganize the amplituhedron into connected contributions to $(AB)_i(AB)_j<0$2, and this geometric rewriting explains why connected negative geometries have only the mild $(AB)_i(AB)_j<0$3 divergence even though the full $(AB)_i(AB)_j<0$4-loop amplitude has $(AB)_i(AB)_j<0$5 divergences. After integrating all but one loop, one obtains an infrared-finite one-variable function $(AB)_i(AB)_j<0$6, interpreted as the normalized null Wilson loop with a single Lagrangian insertion,

$(AB)_i(AB)_j<0$7

with

$(AB)_i(AB)_j<0$8

In this framework, ladders are not a separate unrelated sector: they are a special subset of tree negative geometries, namely single-chain trees (2112.06956).

The tree sector obeys a nonlinear differential equation, but the ladder sector simplifies to a linear equation,

$(AB)_i(AB)_j<0$9

with solution

$G$0

The corresponding contribution to the cusp anomalous dimension is

$G$1

This solvability is one reason ladders serve as a benchmark sector for the negative-geometry program (2112.06956).

The same analysis also clarifies the limits of the ladder approximation. The ladder solution is exponentially suppressed and oscillatory at large $G$2, whereas the more general tree truncation gives a much better qualitative approximation to the known strong-coupling behavior. This does not make the ladder sector anomalous; rather, it locates ladders within a hierarchy of solvable negative-geometry truncations, where they are simpler than general trees but less representative of the full strong-coupling regime (2112.06956).

4. Ladder-type negative geometries in ABJM theory

In ABJM theory, ladder-type negative geometries form an all-loop family that can be integrated recursively by Mellin-space methods. The four-point all-loop integrand is built from connected negative geometries on a symplectic slice with conditions

$G$3

After integrating out all but one loop momentum, the remaining finite object depends on the dual-conformal cross ratio

$G$4

For the ladder family, the Mellin representation

$G$5

turns the recursion for adding rungs into an integral transform with gamma-function kernels. A central structural relation is

$G$6

presented as a recursive defining property of the even-loop ladder family (Li, 2024).

This recursion produces finite polylogarithmic functions at intermediate stages. The paper states that $G$7 is a simple constant, $G$8 is a weight-2 polylogarithm, $G$9 is a polynomial in $\log M$0, and higher even ladders are generated recursively from lower ones. It also emphasizes a distinction between intermediate integrated functions and the final cusp anomalous dimension: the functions $\log M$1 do contain zeta values such as $\log M$2 and $\log M$3, but these do not survive in the cusp coefficient. The final ladder contribution is

$\log M$4

which contains only powers of $\log M$5 (Li, 2024).

A complementary four-loop study computes the infrared-finite functions obtained by integrating three loops of the four-loop negative-geometry integrand. There, the canonical form decomposes as

$\log M$6

showing that ladders coexist with star and box topologies once $\log M$7. The same work finds that for $\log M$8 only ladder diagrams contribute, and that at four loops the box contribution is weaker in the infrared than the others. The extracted four-loop cusp anomalous dimension is

$\log M$9

in agreement with the integrability-based proposal, and the paper also reports an apparent alternating sign pattern for integrated negative geometries in the Euclidean region (Lagares et al., 2024).

5. Leading singularities, Landau analysis, and symbol alphabets

The singularity structure of ladder negative geometries has been analyzed by combining maximal-codimension boundaries of the geometry with geometric Landau analysis. For ladder geometries, the integrated object is decomposed as

$_n$0

where the $_n$1 are one-loop MHV amplituhedron canonical forms with one extra constraint $_n$2, and the $_n$3 are pure functions of weight $_n$4. The underlying leading singularities are classified recursively for arbitrary loop order, using the fact that each loop line lives in $_n$5 and is localized by cut conditions such as

$_n$6

This yields a complete ladder leading-singularity classification (Chicherin et al., 7 Aug 2025).

The same work formulates a geometric criterion separating physical from spurious Landau singularities. One first associates Landau diagrams to maximal boundaries, then pinches propagators to obtain subleading diagrams, and finally discards those whose associated variety does not intersect the positive geometry in maximal real dimension. For negative geometries, the paper refines the rule further by requiring that subleading diagrams with helicity weight $_n$7 be discarded when they arise from a leading Landau diagram of minimal helicity weight $_n$8. This geometric selection rule is essential because ordinary Landau analysis is blind to numerators (Chicherin et al., 7 Aug 2025).

At two loops, the resulting singularity classes at all multiplicities include simple rational letters such as

$_n$9

determinant-type square-root letters, mixed bracket expressions, and five-point Gram determinants built from bi-twistors. After specialization, the five-point two-loop ladder alphabet is exactly the $y_a$0-letter planar pentagon alphabet, while the six-point two-loop ladder alphabet has $y_a$1 independent letters, with $y_a$2 parity-even and $y_a$3 parity-odd. At five points and three loops, the conjectural alphabet has $y_a$4 letters, obtained by adjoining $y_a$5 new letters to the $y_a$6 planar two-loop pentagon letters (Chicherin et al., 7 Aug 2025).

These results define the ladder sector sharply enough to compare it with the first genuinely cyclic case beyond ladders. For four-point one-cycle negative geometries, recursive geometric Landau analysis proves that the integrated function has branch-point singularities only at

$y_a$7

to all loop orders. The paper explicitly identifies one-cycle geometries as the next-to-leading terms in the expansion over cycle number and describes them as the first genuinely cyclic case beyond ladders (Paranjape et al., 24 Apr 2026).

6. Relation to trees, positive ladders, and cyclic sectors

The phrase “ladder negative geometry” is used in more than one precise sense, and the distinctions matter. In four-point planar $y_a$8 SYM, ladders are a special subset of tree negative geometries, defined as single-chain trees. In arbitrary-multiplicity loop space, however, one has both negative ladders and positive ladders, both described uniformly in terms of chamber decompositions and maximal cuts. In ABJM, ladder-type negative geometries are the simplest all-loop bipartite family, but at four loops they no longer exhaust the full negative-geometry decomposition because star and box topologies also appear (2112.06956, Glew et al., 2024, Lagares et al., 2024).

A common misconception is to treat ladder negative geometries as ordinary ladder diagrams. The literature states otherwise. These objects are geometric building blocks: their canonical forms are associated with positive or negative geometry boundaries, and their integrands can be non-planar in dual space even though the final full observable is physical. Likewise, the connected-graph expansion for $y_a$9 is not a Feynman-diagram expansion; it is an expansion in canonical forms of connected negative geometries (2112.06956, Chicherin et al., 7 Aug 2025).

Another recurring misconception is that ladders define the entire solvable sector. The papers instead place them inside a broader hierarchy. Trees are more general and obey a nonlinear equation at four points, ladders obey a simpler linear equation, and one-cycle geometries are the first genuinely cyclic extension beyond ladders. This suggests that ladder negative geometries occupy an intermediate position: they are simple enough to admit all-loop formulas, but structured enough to expose the chamber decomposition, maximal-cut organization, and singularity constraints that continue to shape more general negative geometries (2112.06956, Paranjape et al., 24 Apr 2026).