Triangulation by On-Shell Diagrams in Scattering Amplitudes

• Triangulation by on-shell diagrams is a method that decomposes global Grassmannian forms into canonical cells, each corresponding to a specific leading singularity of an amplitude.
• It employs combinatorial techniques, including BCFW bridges and R-operators, to map diagram edge variables to matrix entries, ensuring a systematic cover of the kinematic space.
• This framework unifies scattering amplitudes, form factors, and polytope structures in both planar and nonplanar supersymmetric Yang–Mills theory using integrability-informed methods.

Triangulation by on-shell diagrams is a central procedure in modern approaches to scattering amplitudes and related structures in supersymmetric gauge theory. It involves decomposing global geometric and combinatorial objects such as Grassmannian or amplituhedron forms into canonical domains (“cells”) associated with on-shell diagrams. Each cell corresponds to a specific leading singularity of an amplitude or form factor, and the union of these cells yields a complete cover (a triangulation) of the kinematic space. This method provides a systematic, combinatorial, and integrability-informed framework that ties together on-shell representations, Grassmannians, and the integrability of planar $\mathcal{N}=4$ supersymmetric Yang–Mills theory (Frassek et al., 2015, Arkani-Hamed et al., 2014, Bai et al., 2014, Farrow et al., 2017, Franco et al., 2013).

1. Grassmannian Formulation and Master Integrals

The foundational structure is the integral over the complex Grassmannian $G(k,n)$ or its extension $G(k,n+2)$, where $k$ is the degree (e.g., MHV, NMHV) and $n$ is the number of external legs. In the context of tree-level form factors of $\mathcal{N}=4$ SYM, two auxiliary on-shell legs encode the off-shell momentum and supermomentum of the operator insertion, leading to a master integral:

$$F_{n,k}(1,\dots,n;q,\gamma^-) = \int\frac{d^{k\times(n+2)}C'}{\mathrm{Vol}[GL(k)]}\;\Omega_{n,k}(C')\; \delta^{2k}(C'\cdot\tilde\lambda)\, \delta^{4k}(C'\cdot\tilde\eta)\, \delta^{2(n+2-k)}(C'^\perp\cdot\lambda)$$

The integrand $\Omega_{n,k}(C')$ encodes a product of consecutive minors and a form factor-specific factor that arises from gluing the minimal form factor. The contour choices and the structure of delta functions select the on-shell loci in kinematic space (Frassek et al., 2015).

2. On-Shell Diagrams and Cells of the Grassmannian

Each term (residue) in the expansion of the Grassmannian integral corresponds to a unique on-shell diagram—a bipartite graph built from trivalent black/white MHV/anti-MHV vertices, minimal form factor vertices, and BCFW bridges (R-operators). These diagrams specify a particular cell in the positive Grassmannian, a domain where all relevant ordered minors are positive. Combinatorially, diagrams are labeled by permutations $\sigma \in S_{n+2}$, encoding the routing of legs through the diagram. The sequence of BCFW bridges attached in reverse order of a permutation decomposition constructs the corresponding cell (Frassek et al., 2015, Franco et al., 2013).

This connection is implemented through a “boundary measurement” that maps edge variables of the diagram to matrix entries in the Grassmannian, thereby specifying a positroid cell or more general matroid cell (for nonplanar cases). The on-shell canonical form pulls back to a $d\log$ measure in these edge variables, corresponding to the canonical differential form on the cell (Franco et al., 2013).

3. Triangulation: Expansions and Complete Covers

The full amplitude or form factor is not given by a single cell, but rather as a sum over residues associated with different on-shell diagrams (cells). This sum constitutes a triangulation of the image of the Grassmannian (or amplituhedron, in twistor variables). The residues are selected by imposing vanishing conditions on minors (e.g., setting specific minors to zero), which geometrically restricts the integration to cells covering the master domain. For example, the NMHV case involves picking residues at $G(k,n)$0; each such residue matches an on-shell diagram missing leg $G(k,n)$1 (Frassek et al., 2015, Farrow et al., 2017).

The triangulation not only guarantees full coverage of the kinematic domain but also realizes efficient constructive and algebraic representations of amplitudes, form factors, and their leading singularities. For nonplanar diagrams or more general polytopes, partial or extended matroid stratifications are invoked to cover the space through multiple diagrammatic representations (Franco et al., 2013, Arkani-Hamed et al., 2014).

4. Canonical Forms, Parke-Taylor Sums, and Polytope Structures

In the MHV sector, individual on-shell diagrams (labeled by sets of triples specifying the black vertices) yield canonical forms that can themselves be expressed as positive sums of Parke–Taylor factors:

$G(k,n)$2

The on-shell form associated with a diagram is

$G(k,n)$3

where the sum is over all orderings consistent with the triple constraints. This decomposition arises directly from the extended positivity conditions on the Grassmannian, with each Parke–Taylor factor associated to a positive cell in a triangulation. More generally, the union of on-shell diagram cells via such expansions provides a cohomological decomposition of the canonical form on the global domain. This underpins polylogarithmic representations of general amplitudes, with the polylog functions weighted by Parke–Taylor factors (Arkani-Hamed et al., 2014).

5. Beyond Planarity, Amplituhedron, and Loop Generalizations

Triangulation by on-shell diagrams is not limited to the planar or MHV setting. In nonplanar amplitudes and higher MHV degree, the combinatorial structure extends through the collection of all reduced diagrams, enabling the construction of (partial) matroid stratifications. For example, in the annulus case (Gr_{3,5} with two boundaries), a full hierarchy of facet moves and partial stratifications constructs the corresponding amplituhedron-like object (Franco et al., 2013).

For loop-level amplitudes, the formalism is adapted by adding loop “bubble” subdiagrams and extracting forward-limit contributions. Each loop adds two rows to the Grassmannian (in twistor space) or a bubble diagram imposing additional cut conditions. Each loop diagram corresponds to a specific cell in the extended positive Grassmannian and provides a triangulation of the multi-loop amplituhedron (Bai et al., 2014, Farrow et al., 2017).

6. Integrability and the Role of R-Operators

BCFW bridges are encoded algebraically via R-operators, which are integral operators acting on pairs of legs with a spectral parameter. These R-operators satisfy the Yang–Baxter equation and thus realize the integrable structure (e.g., Yangian symmetry) underlying the diagrammatic expansion. The transfer matrix constructed from the Lax operators acts as a symmetry generator; for amplitudes this symmetry is exact, while for form factors, Yangian invariance is broken only by the operator insertion but persists on the on-shell parts. The triangulation process respects these integrability structures at the combinatorial and algebraic level, with the sum over on-shell diagrams yielding invariance under the homogeneous transfer matrix (Frassek et al., 2015).

7. Special Instance: Multi-Box Triangulation in Six Dimensions

An explicit variant of triangulation occurs in the analytic evaluation of multi-box Feynman diagrams in six dimensions. Here, the star–triangle relation allows the reduction of box ladder diagrams to a sequence of triangle integrals, trading complicated loop-momentum integrations for purely parametric $G(k,n)$4-fold integrals over Feynman parameters—this process is also described as “triangulation.” The resulting representation is efficient for numerical evaluation and exhibits a recursive structure through which each additional loop introduces two new parameters and modifies the integration polynomial in a fixed pattern (Kazakov, 2014).

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Fundamentally, triangulation by on-shell diagrams provides a rigorous, constructive, and integrability-compatible decomposition of both tree and loop-level scattering amplitudes, form factors, and related geometric objects such as the amplituhedron. This approach connects the combinatorics of diagrams, the geometry of Grassmannians, and the algebra of integrability into a unified framework that underlies much of modern amplitude theory (Frassek et al., 2015, Franco et al., 2013, Arkani-Hamed et al., 2014, Bai et al., 2014, Farrow et al., 2017, Kazakov, 2014).