Graph-Based Hidden Zeros in Scattering and Wavefunctions

• Graph-based hidden zeros are nontrivial vanishing loci defined by graph structures that regulate amplitudes, wavefunctions, and partition functions.
• They unify domains such as scattering amplitudes, cosmological wavefunctions, and matrix theories by utilizing graph orderings, tubings, and clique decompositions.
• This framework provides novel insights into recursion relations, unitarity, and zero-free regions, paving the way for advanced theoretical and computational research.

to=arxiv_search.search 天天中彩票篮球json 天天彩票网 天天中彩票派奖 "{\"query\":\"all:(\\"hidden zeros\" graph amplitudes cosmological wavefunction Ising copositive) OR ti:\\"Hidden Zeros\" OR ti:\\"Motion of Lee-Yang zeros\"\", \"max_results\": 10, \"sort_by\": \"submittedDate\", \"sort_order\": \"descending\"}" to=arxiv_search.search 娱乐平台招商 ฝ่ายขายรายการjson 山大发ුඹ "{\"query\":\"ti:\\"Hidden Zeros\" OR ti:\\"Motion of Lee-Yang zeros\" OR ti:\\"A near-optimal zero-free disk for the Ising model\" OR ti:\\"Representation of Zeros of a Copositive Matrix via Maximal Cliques of a Graph\"\", \"max_results\": 20, \"sort_by\": \"submittedDate\", \"sort_order\": \"descending\"}" Graph-based hidden zeros arise when a graph, ordering, tubing, clique structure, or sparsity pattern determines nontrivial vanishing loci of amplitudes, wavefunctions, partition functions, or matrix polynomials. In scattering amplitudes and cosmological wavefunctions, the term refers to finite kinematic loci where quantities vanish away from standard factorization poles; in statistical mechanics it refers to partition-function zeros whose geometry is constrained by the interaction graph; and in matrix and zeon settings it refers to zero sets organized by maximal cliques, prescribed graph sparsity, or graph-encoded nilpotent structure (Huang et al., 11 Feb 2025, De et al., 30 Mar 2025, Hou et al., 2022, Kostyukova et al., 2024, Drton et al., 2010, Staples, 2021).

1. Scope and core constructions

Several technically distinct notions fall under the same descriptive label because the vanishing mechanism is encoded by graph data rather than by a single universal algebraic definition.

Domain	Graph object	Zero notion
Scattering amplitudes	Planar cubic graphs, causal diamonds, compatible orderings	Vanishing of $A_n$ or $m(\alpha\mid\beta)$ on loci $s_{ab}=0$ across a partition
Cosmological wavefunctions	Graph tubings, graph associahedra, generating graphs	Wavefunction, factorization, and parametric zeros of $\psi_G$ or $\widetilde\psi_G$
Statistical mechanics	Interaction graph, complete graph, bounded-degree graph	Lee–Yang and Fisher zeros; zero-free disks; non-crossing trajectories
Matrix and algebraic problems	Maximal-clique graph, sparsity graph, zeon basis encoding	Convex unions of zeros, hidden semialgebraic constraints, spectrally simple zeon zeros

In the amplitude literature, the basic kinematic setup picks two distinguished legs $i$ and $j$, partitions the remaining legs into two non-empty sets $A$ and $B$, and imposes

$$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$

For BAS double-partial amplitudes, if both orderings are compatible with $m(\alpha\mid\beta)$0, then the amplitude vanishes on this locus (Huang et al., 11 Feb 2025). In cosmological wavefunctions, the relevant graph data are tubings and tube energies $m(\alpha\mid\beta)$1, and the hidden zeros are linear loci in these variables rather than ordinary pole conditions (De et al., 30 Mar 2025). In copositive matrix theory, zeros are vectors $m(\alpha\mid\beta)$2 with $m(\alpha\mid\beta)$3, and a graph built from pairwise $m(\alpha\mid\beta)$4-orthogonality of minimal zeros determines the full zero set (Kostyukova et al., 2024).

2. Tree-level amplitudes: compatible orderings, causal diamonds, and universal expansions

For bi-adjoint scalar theory, the double-partial amplitude is

$m(\alpha\mid\beta)$5

where $m(\alpha\mid\beta)$6 is the set of cubic trees simultaneously planar in the orderings $m(\alpha\mid\beta)$7 and $m(\alpha\mid\beta)$8. When $m(\alpha\mid\beta)$9 and $s_{ab}=0$0 are both compatible with the partition $s_{ab}=0$1, the graph sum vanishes on $s_{ab}=0$2 for all cross pairs. The graph-theoretic mechanism is an $s_{ab}=0$3–$s_{ab}=0$4 chain identity: once the attached $s_{ab}=0$5-blocks and $s_{ab}=0$6-blocks are fixed, summing over their allowed interleavings gives a telescoping expression proportional to an on-shell invariant that vanishes (Huang et al., 11 Feb 2025).

This BAS zero propagates to other theories through universal expansions. Yang–Mills, the NLSM, special Galileon, Born–Infeld, and gravity can all be expanded in BAS double-partials with kinematic coefficients. Representative formulas are

$s_{ab}=0$7

$s_{ab}=0$8

and

$s_{ab}=0$9

Kleiss–Kuijf rearrangements convert shuffle sums into BAS terms with orderings compatible with the partition, so the BAS zero implies the vanishing of the parent amplitude (Huang et al., 11 Feb 2025, Zhou, 13 Oct 2025).

A complementary formulation uses causal diamonds in the kinematic mesh. At six points, the relevant loci include height-2 “squares” and height-1 “skinny rectangles,” defined by $\psi_G$0 for selected disjoint subsets $\psi_G$1. In scaffolded General Relativity, multi-flavor DBI, and special Galileon, these loci produce vanishing amplitudes and near-zero factorizations with prefactors built from planar variables $\psi_G$2 and $\psi_G$3. In the scaffolded GR case, the near-zero limits select a metric-coupled cubic scalar sector with

$\psi_G$4

The same causal-diamond logic extends beyond color ordering to flavor orderings represented as perfect matchings on the external-leg graph (Li et al., 2024).

BCJ and KLT relations provide a parallel route. Hidden zeros in $\psi_G$5, NLSM, and YM follow from BCJ relations, while KLT transports them to permutation-invariant theories such as the special Galileon. At six points, the loci

$\psi_G$6

support both vanishing and near-zero factorization into lower-point objects (Bartsch et al., 2024).

3. Recursion, 2-split behavior, one-loop extensions, and deformations

A major development is the identification of hidden zeros with enhanced large-$\psi_G$7 behavior under BCFW-like deformations. For ordered rational functions built from planar variables $\psi_G$8, each $\psi_G$9-zero is equivalent to a corresponding subset-enhanced scaling statement: if $\widetilde\psi_G$0 is decomposed by large-$\widetilde\psi_G$1 degree, then

$\widetilde\psi_G$2

For $\widetilde\psi_G$3, this equivalence is used to prove that imposing $\widetilde\psi_G$4 distinct 1-zeros determines all planar cubic-graph coefficients up to an overall normalization (Rodina, 2024).

In the NLSM, standard BCFW recursion is obstructed by poor large-$\widetilde\psi_G$5 behavior, but a modified contour cures this by inserting $\widetilde\psi_G$6 and arranging hidden zeros at $\widetilde\psi_G$7. The resulting recursion relation is

$\widetilde\psi_G$8

with only physical factorization poles contributing. The same framework reproduces the Adler zero, the $\widetilde\psi_G$9-shift construction from $i$0, and the universal expansion into bi-adjoint scalar amplitudes (Li et al., 18 Aug 2025). A related modified BCFW analysis proves hidden zeros directly and derives the associated 2-split behavior in terms of carefully defined off-shell currents (Feng et al., 19 Apr 2025).

At one loop, hidden zeros are formulated on the punctured-disk kinematic mesh by “big mountains,” maximal triangular loci of vanishing $i$1-variables. In $i$2, one-loop surface integrands are unitary if and only if they satisfy the loop hidden zeros, assuming locality. Near a loop zero,

$i$3

and the same factorization pattern motivates a conjectural determination of one-loop NLSM integrands without assuming locality or unitarity (Backus et al., 5 Mar 2025).

The zero structure is also stable under some deformations and unstable under others. Uniform-mass $i$4, Kaluza–Klein reductions, spurion-induced massive NLSM, and spontaneously broken gauge theories preserve hidden zeros after replacing $i$5 by mass-deformed $i$6. By contrast, a naive pion mass term in the NLSM and a simple massive Yang–Mills theory spoil the zeros (González et al., 23 Jan 2026). Higher-derivative tree amplitudes also preserve the phenomenon: a single $i$7 insertion in YM and the $i$8 and $i$9 sectors of gravity continue to exhibit hidden zeros, with the unordered gravitational case requiring a detailed $j$0-counting argument to cancel potential propagator singularities (Zhou, 13 Oct 2025).

4. Cosmological wavefunctions, graph tubings, and graph associahedra

For a scalar theory of conformally coupled massless fields with cubic interaction in flat FRW slicing, a graph contribution to the wavefunction is built from tubings: $j$1 The stripped contribution $j$2 removes the universal single-site and total-energy tubes. Within this framework, three kinds of hidden zeros are distinguished. Wavefunction zeros are linear loci causing $j$3 and $j$4 to vanish and occur only for tree chain graphs. Factorization zeros arise after setting an interior-site parameter $j$5 and then imposing the chain wavefunction-zero conditions on the factorized subgraphs. Parametric zeros are obtained by setting parameters such as $j$6 and certain $j$7 to zero; geometrically they are flattenings of the cosmological graph associahedron (De et al., 30 Mar 2025).

The geometry is explicit. The cosmological graph associahedron $j$8 has facet hyperplanes $j$9, complete tubings correspond to vertices, and the cosmological limit $A$0 imposes the linear tube relations

$A$1

Parametric zeros are dimension-dropping flattenings of $A$2, whereas wavefunction and factorization zeros arise from factorizations of the adjoint polynomial in the associated hyperplane arrangement (De et al., 30 Mar 2025).

Chain graphs admit a second interpretation. Under the identification $A$3, the stripped chain wavefunction equals the $A$4-point color-ordered $A$5 amplitude term by term: $A$6 Accordingly, cosmological hidden zeros map to the hidden zeros of colored amplitudes (De et al., 30 Mar 2025).

A later generalization introduces the generating graph $A$7, maps tube variables to Mandelstam invariants $A$8, and defines an amplitude-like object $A$9. The new “blob zero” sets

$B$0

where a boundary pair $B$1 splits the generating graph into two pieces. The core statement is a dual factorization principle: $B$2 Near the zero, each shuffle block collapses into an ordinary product with a universal prefactor, and locality together with the full set of hidden zeros uniquely fixes tree-level cosmological wavefunctions without assuming unitarity (Li et al., 1 Apr 2026).

5. Statistical mechanics: Lee–Yang, Fisher, and zero-free regions on graphs

In the Ising model with ferromagnetic pair couplings on a finite graph $B$3, the partition function

$B$4

has Lee–Yang zeros on the imaginary $B$5-axis, equivalently on the unit circle in the fugacity plane, when the interactions are ferromagnetic. If the subgraph of strictly positive couplings is connected, all Lee–Yang zeros are simple, vary analytically with coupling parameters, and remain strictly ordered. Under variation of a single edge coupling $B$6, each principal zero is either constant, strictly decreasing, or strictly increasing, and its entire trajectory stays within its neighboring interval determined at $B$7. In particular, the trajectories of distinct zeros are disjoint (Hou et al., 2022).

For the Curie–Weiss model on the complete graph, the partition function

$B$8

satisfies the backward heat equation $B$9. All principal zeros are simple for $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$0, and every principal zero decreases strictly in $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$1, converging to $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$2 as $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$3 (Hou et al., 2022).

On annealed scale-free networks, the zero geometry changes. For $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$4, the Lee–Yang zeros reproduce the complete-graph behavior and are purely imaginary. For $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$5, the Fisher zeros hit the real temperature axis at the $$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$6-dependent angle

$$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$7

and the Lee–Yang circle theorem is violated: beyond a finite number of purely imaginary zeros, higher zeros acquire nonzero real part (Krasnytska et al., 2015). The paper explicitly describes these off-axis zeros as revealing a difference between complete graphs and annealed scale-free networks.

A distinct bounded-degree result concerns zero-free regions. Writing the Ising partition function as

$$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$8

one obtains the even-subgraph expansion

$$s_{ab}=(k_a+k_b)^2=0 \qquad \text{for all } a\in A,\ b\in B.$$9

For graphs of maximum degree at most $m(\alpha\mid\beta)$00,

$m(\alpha\mid\beta)$01

and for large girth the radius approaches $m(\alpha\mid\beta)$02. This scale is essentially optimal under a complexity-theoretic assumption (Patel et al., 2023).

6. Matrix, sparsity, and algebraic formulations

For a copositive matrix $m(\alpha\mid\beta)$03, the normalized zero set is

$m(\alpha\mid\beta)$04

Minimal zeros are those with inclusion-minimal support. Writing the finite family of normalized minimal zeros as $m(\alpha\mid\beta)$05, one defines the minimal zeros graph $m(\alpha\mid\beta)$06 by

$m(\alpha\mid\beta)$07

where $m(\alpha\mid\beta)$08. If $m(\alpha\mid\beta)$09 are the maximal cliques of $m(\alpha\mid\beta)$10, then

$m(\alpha\mid\beta)$11

Hidden zeros in this setting are exactly the non-minimal convex combinations supported on maximal cliques with at least two vertices (Kostyukova et al., 2024).

A different graph-theoretic zero problem arises for positive semidefinite matrices with prescribed zeros. Given a graph $m(\alpha\mid\beta)$12, the cone

$m(\alpha\mid\beta)$13

is parametrized by a simplicial complex $m(\alpha\mid\beta)$14 supported on cliques of $m(\alpha\mid\beta)$15 through

$m(\alpha\mid\beta)$16

If $m(\alpha\mid\beta)$17 is chordal and $m(\alpha\mid\beta)$18 is the clique complex, $m(\alpha\mid\beta)$19 is surjective onto $m(\alpha\mid\beta)$20. For non-chordal graphs, especially chordless cycles, the image is strictly smaller than $m(\alpha\mid\beta)$21, and additional semialgebraic inequalities appear. For the cycle $m(\alpha\mid\beta)$22, the exact criterion is

$m(\alpha\mid\beta)$23

These extra constraints are described as hidden relations induced by the latent parametrization rather than by the explicit graph-imposed zeros (Drton et al., 2010).

In zeon algebra, graph-encoded nilpotent coefficients lead to another notion of hidden zero structure. For a monic zeon polynomial $m(\alpha\mid\beta)$24 with scalar part $m(\alpha\mid\beta)$25, a simple zero $m(\alpha\mid\beta)$26 of $m(\alpha\mid\beta)$27 lifts to a unique zeon zero

$m(\alpha\mid\beta)$28

where the grade-$m(\alpha\mid\beta)$29 corrections satisfy

$m(\alpha\mid\beta)$30

The scalar root is visible, while the nilpotent corrections are determined grade by grade by the graph-encoded zeon coefficients (Staples, 2021).

7. Uniqueness, partial-wave realizations, and remaining obstructions

Hidden zeros increasingly function as bootstrap data rather than as isolated vanishing statements. In tree-level NLSM, hidden zeros together with standard factorization uniquely determine all amplitudes (Li et al., 18 Aug 2025). In cosmological wavefunctions, locality plus the full set of blob zeros uniquely fixes tree-level wavefunction coefficients, and the blob-zero conditions are equivalent to enhanced large-$m(\alpha\mid\beta)$31 BCFW scaling (Li et al., 1 Apr 2026). At one loop in $m(\alpha\mid\beta)$32, hidden zeros are equivalent to unitarity for local surface integrands, and there is evidence that locality itself is already encoded by the zeros (Backus et al., 5 Mar 2025).

A partial-wave version appears at five points. For residues of planar-ordered five-point amplitudes with identical scalar external states, imposing two independent splitting loci yields linear relations among the partial-wave coefficients $m(\alpha\mid\beta)$33. At low mass levels these constraints, together with spin truncation, determine the residue in terms of four-point data. Imposing both loci also forces the residue to vanish on their intersection, making the associated hidden zero explicit in partial-wave space. When both channels allow spin-2 exchange, however, a genuine kernel remains in the $m(\alpha\mid\beta)$34 sector, so additional higher-point input is required for complete rigidity (Saha et al., 21 Jan 2026).

These developments suggest a common pattern. Hidden zeros are strongest when the graph combinatorics rigidly organizes allowed propagators, tubings, or clique data, and weakest when degeneracies or large kernels survive. A plausible implication is that graph-based hidden zeros are most informative when they can be paired with a second structural principle—such as locality, factorization on physical poles, maximal-clique decomposition, or strict ordering of zeros—to remove residual ambiguity.