Exact CHY Integrand Construction Using Combinatorial Neural Networks and Discrete Optimization

TL;DR

This work tackles the inverse CHY problem by encoding the pole structure of tree-level amplitudes in a generalized pole degree $K(A)$, defined as a signed edge-count on a colored CHY graph. The authors show additivity under multiplication and derive an elementary face recursion $K(A)=rac{1}{|A|-2}\sum K(B)$, which reduces higher-order pole data to linear functions of the two-particle inputs $K(s_{ij})$, enabling a mixed-integer linear feasibility formulation. A fixed subset lattice graph supports deterministic, parameter-free forward evaluation and backward residual propagation, with a factorial rescaling $ ilde{K}(A)=(|A|-2)!\,K(A)$ ensuring integral updates and exact propagation. The framework introduces a graded, $n$-regular perspective on integrand graphs, highlighting 0-regular (Möbius-invariant) factors that decompose into four-point cross ratios to facilitate pick-pole selection and higher-order-pole reductions, demonstrated in six- and eight-point examples. Overall, the method yields exact, training-free construction of CHY integrands from prescribed pole data via MILP and discrete combinatorial optimization, with scalable pole-reduction techniques via cross-ratio identities and 0-regular graph decompositions.

Abstract

Constructing a rational CHY integrand that realizes prescribed physical pole constraints is a discrete inverse problem whose combinatorial complexity grows with multiplicity. We encode the pole hierarchy through generalized pole degrees $K(A)$ (channels $s_A$), defined as signed internal-edge counts associated with particle subsets in a colored integrand graph. Additivity under integrand multiplication together with the elementary face recursion on the subset lattice expresses all higher-channel $K(A)$ as linear functions of the two-particle data $\{K(s_{ij})\}$ and reduces the inverse step to a mixed-integer linear feasibility problem. The subset lattice provides a fixed dependency graph for deterministic message passing with forward evaluation and backward residual propagation; this computation is parameter-free and involves no training. In factorial-rescaled variables $\widetilde K(A)=(|A|-2)!\,K(A)$, every local update is integral, so propagation is exact in the rescaled recursion variables and does not rely on numerical reconstruction. We further organize generalized integrand graphs by an $n$-regular grading under multiplication, where degree-zero (0-regular) factors act as Möbius-invariant insertions that can be decomposed into four-point cross ratios. We illustrate the construction at six and eight points, including pick-pole selection and higher-order pole reduction.

Figures (39)

• Figure 1: The 4-regular graph of integrand $\mathcal{I} = \text{PT}(123456)\text{PT}(125436)$
• Figure 2: The set of 0-regular graphs is closed under multiplication.
• Figure 3: The $K_{I^{[3]}}(s_A)=K_{I^{[1]}}(s_A)+K_{I^{[2]}}(s_A)$ for all poles
• Figure 4: The "0-regular group" acts as a set of transformations on other graphs
• Figure 5: The $K_{I^{[6]}}(s_A)=K_{I^{[4]}}(s_A)+K_{I^{[5]}}(s_A)$ for all poles

Theorems & Definitions (6)

• Lemma C.1: Alternating cycle decomposition
• proof
• Lemma C.2: Even-cycle telescoping into four-point cross-ratios
• proof
• Theorem C.3: 0-regular decomposition into four-point cross-ratios
• proof