Set of Integrands: Structure and Applications

• Set of integrands are collections of functions or distributions defined on specific domains under algebraic, topological, or combinatorial constraints, forming the basis for integration in mathematics and physics.
• They are constructed via graph theory, computational algebraic geometry, and algorithmic decomposition, which provide universal bases for amplitude expansion and canonical integral evaluation.
• Applications include quantum field theory, scattering amplitude reconstruction, and convex analysis, enabling improved identification of master integrals and deeper insights into complex systems.

A set of integrands, in the modern mathematical and physical literature, refers to a collection of functions (or distributions) defined for integration over domain(s) relevant to a given context—such as spheres, configuration spaces, loop-momentum spaces, algebraic varieties, or phase space—subject to structural, topological, algebraic, or combinatorial constraints. The explicit characterization and construction of such sets underpin fundamental advances in quantum field theory, algebraic geometry, and the analysis of convex functions. Recent research establishes universal bases for integrands in gauge theory, gravity, scattering amplitudes, and convex geometry, organizing their structure by graph theory, algebraic varieties, stability criteria, and symmetry properties.

1. Structural Definition and Topological Classification

A central instance of integrand sets appears in the study of convex integrands. For a sphere $S^n \subset \mathbb{R}^{n+1}$, a $C^\infty$ convex integrand $y: S^n \to \mathbb{R}_+$ is defined via the graph $$\text{graph}(y) = \{ (\theta, y(\theta)) \mid \theta \in S^n \}$$ in polar coordinates and the inversion map $\text{inv}(\theta, r) = (-\theta, 1/r)$, with $W = \text{Conv}(\text{inv}(\text{graph}(y)))$ denoting the convex hull of the inverted graph. The set $\mathcal{C} = C_{\text{conv}}(S^n, \mathbb{R}_+)$ consists of all $C^\infty$ convex integrands, endowed with the Whitney topology, and is stratified by $A$-equivalence classes. The subset of stable integrands $\mathcal{S}$, for which Morse nondegeneracy and uniqueness of critical values hold, is open and dense in $\mathcal{C}$ (Batista et al., 2016).

2. Universal Bases via Graphs and Algebraic Geometry

Modern amplitude construction utilizes universal sets of integrands organized via graph-theoretic and algebraic-geometric methods. For multi-loop quantum field theories, the basis comprises integrands associated to representative graphs (planar and nonplanar), characterized by propagator powers and irreducible scalar products (ISPs). Each basis element corresponds to an integral

$$I^{(\Gamma)}_{a_1,\ldots,a_{E_\Gamma};\,b_1,\ldots,b_{p_\Gamma}} = \int \prod_{i=1}^L \frac{d^D\ell_i}{(2\pi)^D} \frac{\sigma_1^{b_1}\cdots \sigma_{p_\Gamma}^{b_{p_\Gamma}}}{\rho_1^{a_1} \cdots \rho_{E_\Gamma}^{a_{E_\Gamma}}},$$

where $\Gamma$ is a cubic graph, $\{\rho_i\}$ are inverse propagators, and $\{\sigma_j\}$ span the ISPs. The full basis is a union over the graph set $G(L, n)$ filtered to eliminate redundancy, yielding a complete and non-overcomplete set for amplitude expansion (Bern et al., 2024).

3. Algorithmic Decomposition and Basis Construction

Integrand sets admit systematic construction through partial-fractioning and algebraic-geometry ideals. In real-emission phase-space integrals at next-to-leading order, the decomposition algorithm involves identifying denominator polynomials in momenta and remapping constraints, forming an ideal $I$, and employing computational algebraic geometry (Gröbner bases, Bézout identities). Integrands with five or more $k_r$-dependent denominators reduce to combinations with four or fewer; thus, the universal basis consists of integrands with at most four dependent denominators, known as "triskelia". This reduction is dimensionally unique and matches to master integrals, enabling analytic integration (Kosower et al., 2022).

4. Integrand Bases in Scattering Equations and Holomorphic Approaches

Loop-level scattering amplitudes are organized via sets of integrands built from CHY (Cachazo-He-Yuan) formalism and holomorphic forms. For scalar and gauge theories, the CHY integrand is constructed by mapping Feynman diagrams to products of quadratic differentials derived from holomorphic forms on nodal curves with punctures representing loop momenta. The graph-theoretic dictionary between cubic graphs and CHY integrands applies up to two loops and extends to higher genus via quadratic differentials and skeleton factors. For bi-adjoint $\phi^3$ theory, the full set of integrands up to seven legs corresponds to analytic test cases of the CHY building block prescription (Gomez et al., 2016). In gauge theory, integrand sets capture color-kinematics duality and match generalized unitarity cuts, transitioning from linear to quadratic propagator forms via double forward limits and prescriptions for matching tree amplitudes (Agerskov et al., 2019).

5. Combinatorial and Geometric Constraints: Generalized Color Orderings

Beyond $k=2$ cases (CHY), for $k>2$ in the CEGM representation, sets of integrands are constructed combinatorially from generalized color orderings (GCOs), defined by arrangements on configuration space $X(k,n)$ of complex $k \times n$ matrices, subject to SL$(k)$ invariance and torus quotient. For each GCO, constraints on pole structure (Plücker minors), torus weights, and simplex-flip residue matching uniquely determine the integrand. Explicit integrand sets for $(3, n)$ and $(4, n)$ have been classified and tabulated. These sets encode combinatorial identities (decoupling and double-extension identities) and admit interpretations via points, hyperplanes, and chirotope signs (Cachazo et al., 2023).

6. Canonical Integrands and Leading Singularities

Sets of integrands serving as solutions to canonical differential equations are selected via leading singularities (LS), particularly in degenerate and elliptic examples. In the genus-zero case, bases of $\mathrm{d}\log$ integrands with integer LS give rise to Feynman integrals satisfying canonical $\epsilon$-differential equations. In genus-one (elliptic) cases, integrand sets are built from algebraic one-forms (first, second, third kind), and normalization cycles fix integrands to have unit LS, resulting in factorized differential equations of the form $d\bar G = [g - n\epsilon] A_0(x) + \epsilon A_1(x)$. The conjectured universality asserts that for any Feynman family with single-elliptic maximal cut, such integrand-level construction yields canonical $\epsilon$-factorized systems of pure weight (Chaubey et al., 29 Apr 2025).

7. Physical, Geometric, and Computational Implications

Analysis of integrand sets clarifies robustness and universality in physical applications:

• For convex integrands on $S^n$, generic choices correspond to Morse functions with distinct critical values; the open-dense property implies that variational problems are generically reducible to stable cases.
• In amplitude construction, the global basis enables direct mapping of unitarity cuts to integrand coefficients, simplifies double copy to gravity, and manifests gauge-parameter independence, on-shell symmetries, and stratification by power counting.
• Integrand sets organized by geometric, combinatorial, or algebraic properties are foundational for future advances in high-loop computations, canonical form theory, and the analysis of special function transcendentality.

These frameworks, developed by groups including Chaubey–Sotnikov, Mizera–Gomez–Zhang, Mastrolia–Sogaard, and many others, underpin current approaches to classifying, constructing, and evaluating integrand sets throughout theoretical physics and mathematics (Batista et al., 2016, Kosower et al., 2022, Gomez et al., 2016, Agerskov et al., 2019, Bourjaily et al., 2020, Bern et al., 2024, Chaubey et al., 29 Apr 2025, Cachazo et al., 2023).