---
title: Graph-Based Hidden Zeros in Scattering and Wavefunctions
url: https://www.emergentmind.com/topics/graph-based-hidden-zeros
type: topic
---

# Graph-Based Hidden Zeros in Scattering and Wavefunctions

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 [2502.07173], [2503.23579], [2208.00917], [2410.08066], [1001.3195], [2103.11227].

## 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 \((A,i,B,j)\), then the amplitude vanishes on this locus [2502.07173]. In cosmological wavefunctions, the relevant graph data are tubings and tube energies \(S_\tau\), and the hidden zeros are linear loci in these variables rather than ordinary pole conditions [2503.23579]. In copositive matrix theory, zeros are vectors \(x\ge 0\) with \(x^T A x=0\), and a graph built from pairwise \(A\)-orthogonality of minimal zeros determines the full zero set [2410.08066].

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

For bi-adjoint scalar theory, the double-partial amplitude is
\[
m_n(\alpha\mid\beta)=\sum_{g\in \mathcal{G}(\alpha)\cap \mathcal{G}(\beta)}\ \prod_{e\in E(g)}\frac{1}{s_e},
\]
where \(\mathcal{G}(\alpha)\cap \mathcal{G}(\beta)\) is the set of cubic trees simultaneously planar in the orderings \(\alpha\) and \(\beta\). When \(\alpha\) and \(\beta\) are both compatible with the partition \((A,i,B,j)\), the graph sum vanishes on \(s_{ab}=0\) for all cross pairs. The graph-theoretic mechanism is an \(i\)–\(j\) chain identity: once the attached \(A\)-blocks and \(B\)-blocks are fixed, summing over their allowed interleavings gives a telescoping expression proportional to an on-shell invariant that vanishes [2502.07173].

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
\[
A_n^{\rm YM}(\boldsymbol{\sigma})=\sum_{\boldsymbol{\pi}} C_{\rm YM}(\epsilon,\boldsymbol{\pi})\, m_n\big(i,\boldsymbol{\pi},j\mid \boldsymbol{\sigma}\big),
\]
\[
A_n^{\rm NLSM}(\boldsymbol{\sigma})=\sum_{\boldsymbol{\pi}} \hat C(\boldsymbol{\pi})\, m_n\big(i,\boldsymbol{\pi},j\mid \boldsymbol{\sigma}\big),
\]
and
\[
M_n^{R^3}=\sum_{\boldsymbol{\pi},\boldsymbol{\pi}'} C_{F^3}(\epsilon,\boldsymbol{\pi})\,C_{F^3}(\tilde\epsilon,\boldsymbol{\pi}')\,
m_n\big(i,\boldsymbol{\pi},j\mid i,\boldsymbol{\pi}',j\big).
\]
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 [2502.07173], [2510.11070].

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 \(s_{\alpha,\beta}=0\) for selected disjoint subsets \(\alpha,\beta\). 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 \(X_T\) and \(X_B\). In the scaffolded GR case, the near-zero limits select a metric-coupled cubic scalar sector with
\[
\mathcal{L}^{\text{ext}} = \sqrt{-g}\left[ -\frac12 g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi + \Phi^3 \right].
\]
The same causal-diamond logic extends beyond color ordering to flavor orderings represented as perfect matchings on the external-leg graph [2403.12939].

BCJ and KLT relations provide a parallel route. Hidden zeros in \(\mathrm{Tr}(\phi^3)\), 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
\[
C_1=\{s_{13}=s_{14}=s_{15}=0\},\qquad
C_2=\{s_{14}=s_{15}=s_{24}=s_{25}=0\}
\]
support both vanishing and near-zero factorization into lower-point objects [2403.10594].

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

A major development is the identification of hidden zeros with enhanced large-\(z\) behavior under BCFW-like deformations. For ordered rational functions built from planar variables \(X_{ij}\), each \(k\)-zero is equivalent to a corresponding subset-enhanced scaling statement: if \(B=B_m+B_{m-1}+\cdots\) is decomposed by large-\(z\) degree, then
\[
B \text{ satisfies the } k\text{-zero}
\iff
B_i(X^{(0)})=0 \text{ for all } i
\iff
B_i(X(z))\sim z^{i-1}.
\]
For \(\mathrm{Tr}(\phi^3)\), this equivalence is used to prove that imposing \(n-3\) distinct 1-zeros determines all planar cubic-graph coefficients up to an overall normalization [2406.04234].

In the NLSM, standard BCFW recursion is obstructed by poor large-\(z\) behavior, but a modified contour cures this by inserting \(1/(1-z^2)\) and arranging hidden zeros at \(z=\pm1\). The resulting recursion relation is
\[
A_{2n}^{\text{NLSM}}
=
\sum_{z^*:~S(z^*)=0}
\frac{
A_{2n_1}^{\text{NLSM}}(z^*)
A_{2n+2-2n_1}^{\text{NLSM}}(z^*)
}{
S\,[\,1-(z^*)^2\,]
},
\]
with only physical factorization poles contributing. The same framework reproduces the Adler zero, the \(\delta\)-shift construction from \(\mathrm{Tr}(\phi^3)\), and the universal expansion into bi-adjoint scalar amplitudes [2508.12894]. A related modified BCFW analysis proves hidden zeros directly and derives the associated 2-split behavior in terms of carefully defined off-shell currents [2504.14215].

At one loop, hidden zeros are formulated on the punctured-disk kinematic mesh by “big mountains,” maximal triangular loci of vanishing \(c\)-variables. In \(\mathrm{Tr}(\phi^3)\), one-loop surface integrands are unitary if and only if they satisfy the loop hidden zeros, assuming locality. Near a loop zero,
\[
\mathcal{I}_n\left(c_{\star} \neq 0\right)=\left( \frac{1}{Y_i^\mp} + \frac{1}{Y_{i-1}^\pm} \right) \times \mathcal{A}_{n+2},
\]
and the same factorization pattern motivates a conjectural determination of one-loop NLSM integrands without assuming locality or unitarity [2503.03805].

The zero structure is also stable under some deformations and unstable under others. Uniform-mass \(\mathrm{Tr}\,\Phi^3\), Kaluza–Klein reductions, spurion-induced massive NLSM, and spontaneously broken gauge theories preserve hidden zeros after replacing \(X_{i,j}\) by mass-deformed \(\widetilde X_{i,j}\). By contrast, a naive pion mass term in the NLSM and a simple massive Yang–Mills theory spoil the zeros [2601.16860]. Higher-derivative tree amplitudes also preserve the phenomenon: a single \(F^3\) insertion in YM and the \(R^2\) and \(R^3\) sectors of gravity continue to exhibit hidden zeros, with the unordered gravitational case requiring a detailed \(\tau\)-counting argument to cancel potential propagator singularities [2510.11070].

## 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:
\[
\psi_G = \sum_{\mathbb{T}} \prod_{\tau\in\mathbb{T}} \frac{1}{S_\tau},\qquad
S_\tau = \sum_{v\in\tau} X_v + \sum_{e \text{ crosses }\tau} Y_e.
\]
The stripped contribution \(\widetilde\psi_G\) 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 \(\psi_G\) and \(\widetilde\psi_G\) to vanish and occur only for tree chain graphs. Factorization zeros arise after setting an interior-site parameter \(S_j\to0\) and then imposing the chain wavefunction-zero conditions on the factorized subgraphs. Parametric zeros are obtained by setting parameters such as \(S_{\rm total}\) and certain \(S_i\) to zero; geometrically they are flattenings of the cosmological graph associahedron [2503.23579].

The geometry is explicit. The cosmological graph associahedron \(\widetilde A_G\) has facet hyperplanes \(S_\tau=0\), complete tubings correspond to vertices, and the cosmological limit \(\delta_\tau\to0\) imposes the linear tube relations
\[
S_{\tau_1}+S_{\tau_2}=S_{\tau_1\cup\tau_2}+S_{\tau_1\cap\tau_2}.
\]
Parametric zeros are dimension-dropping flattenings of \(\widetilde A_G\), whereas wavefunction and factorization zeros arise from factorizations of the adjoint polynomial in the associated hyperplane arrangement [2503.23579].

Chain graphs admit a second interpretation. Under the identification \(S_{i\cdots j-1}\equiv X_{i,j}\), the stripped chain wavefunction equals the \((n+1)\)-point color-ordered \(\mathrm{Tr}(\phi^3)\) amplitude term by term:
\[
\widetilde{\psi}_{n\text{-chain}}(S)\equiv A_{n+1}^{\mathrm{Tr}(\phi^3)}(X).
\]
Accordingly, cosmological hidden zeros map to the hidden zeros of colored amplitudes [2503.23579].

A later generalization introduces the generating graph \(G=\mathcal G\cup\{v_{n+1}\}\), maps tube variables to Mandelstam invariants \(S_I\mapsto s_I\), and defines an amplitude-like object \(\mathcal A_G\). The new “blob zero” sets
\[
p_{a\in A}\cdot p_{b\in B}=0 \qquad \text{for all } (a,b)\in A\times B,
\]
where a boundary pair \(\{i,j\}\) splits the generating graph into two pieces. The core statement is a dual factorization principle:
\[
\mathcal A_G \text{ satisfies the blob zero on } (A,B)
\quad\Longleftrightarrow\quad
\mathcal A_G=\mathcal A_{G_A}\shuffle \mathcal A_{G_B}.
\]
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 [2604.01133].

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

In the Ising model with ferromagnetic pair couplings on a finite graph \(G=(V,E)\), the partition function
\[
Z_G(h;J) = \sum_{\sigma\in\{\pm1\}^V}
\exp\!\left(
\sum_{uv\in E} J_{uv}\sigma_u\sigma_v + h\sum_{u\in V}\sigma_u
\right)
\]
has Lee–Yang zeros on the imaginary \(h\)-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 \(t\), each principal zero is either constant, strictly decreasing, or strictly increasing, and its entire trajectory stays within its neighboring interval determined at \(t=0\). In particular, the trajectories of distinct zeros are disjoint [2208.00917].

For the Curie–Weiss model on the complete graph, the partition function
\[
Z_{n,t}(x) := \sum_{\sigma\in\{\pm1\}^n} \exp\!\left( t \Big[\sum_{j=1}^n \sigma_j\Big]^2 + i x \sum_{j=1}^n \sigma_j \right)
\]
satisfies the backward heat equation \(\partial_t Z_t(x)=-\partial_{xx}Z_t(x)\). All principal zeros are simple for \(t>0\), and every principal zero decreases strictly in \(t\), converging to \((2k-1)\pi/(2n)\) as \(t\to\infty\) [2208.00917].

On annealed scale-free networks, the zero geometry changes. For \(\lambda>5\), the Lee–Yang zeros reproduce the complete-graph behavior and are purely imaginary. For \(3<\lambda<5\), the Fisher zeros hit the real temperature axis at the \(\lambda\)-dependent angle
\[
\phi(\lambda)=\frac{\pi(\lambda-3)}{2(\lambda-1)},
\]
and the Lee–Yang circle theorem is violated: beyond a finite number of purely imaginary zeros, higher zeros acquire nonzero real part [1510.00534]. 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
\[
Z_{\rm Ising}(G;b)=\sum_{\sigma:V\to\{0,1\}} b^{m(\sigma)},
\qquad
x=\frac{b-1}{b+1},
\]
one obtains the even-subgraph expansion
\[
Z_{\rm even}(G;x)=\sum_{F\in \mathrm{Even}(G)} x^{|F|}.
\]
For graphs of maximum degree at most \(\Delta\),
\[
Z_{\rm Ising}(G;b)\neq 0
\quad\text{whenever}\quad
\left|\frac{b-1}{b+1}\right|
\le
\frac{\left(1-\frac{1}{\sqrt{2(\Delta-1)}}\right)^2}{\Delta-1}
=
\frac{1-o_\Delta(1)}{\Delta-1},
\]
and for large girth the radius approaches \((1-\varepsilon)/(\Delta-1)\). This scale is essentially optimal under a complexity-theoretic assumption [2311.05574].

## 6. Matrix, sparsity, and algebraic formulations

For a copositive matrix \(A\in C^n\), the normalized zero set is
\[
Z_{\rm norm}(A):=\{x\in\mathbb R^n:\ x\ge0,\ \|x\|_1=1,\ x^TAx=0\}.
\]
Minimal zeros are those with inclusion-minimal support. Writing the finite family of normalized minimal zeros as \(\{T(j):j\in J\}\), one defines the minimal zeros graph \(G(A)\) by
\[
(i,j)\in V
\iff
\operatorname{supp}(T(i))\subseteq M(j)
\iff
T(i)^TAT(j)=0,
\]
where \(M(j)=\{k:e_k^TAT(j)=0\}\). If \(\{J(s):s\in S\}\) are the maximal cliques of \(G(A)\), then
\[
Z_{\rm norm}(A)=\bigcup_{s\in S} \operatorname{conv}\{T(j): j\in J(s)\}.
\]
Hidden zeros in this setting are exactly the non-minimal convex combinations supported on maximal cliques with at least two vertices [2410.08066].

A different graph-theoretic zero problem arises for positive semidefinite matrices with prescribed zeros. Given a graph \(G\), the cone
\[
S_+^G=\{\Sigma\in S_+^p:\ \Sigma_{ij}=0 \text{ for all } i\neq j \text{ and } \{i,j\}\notin E\}
\]
is parametrized by a simplicial complex \(K\) supported on cliques of \(G\) through
\[
\Phi_K(\theta)=\Gamma(\theta)\Gamma(\theta)^{\mathsf T}.
\]
If \(G\) is chordal and \(K\) is the clique complex, \(\Phi_K\) is surjective onto \(S_+^G\). For non-chordal graphs, especially chordless cycles, the image is strictly smaller than \(S_+^G\), and additional semialgebraic inequalities appear. For the cycle \(C_m\), the exact criterion is
\[
\sum_{M\in\mathcal{M}(C_m)} (-1)^{|M|}
\Bigl(\prod_{\{i,j\}\in M} \sigma_{ij}^2\Bigr)
\Bigl(\prod_{i\in [m]\setminus M} \sigma_{ii}\Bigr)
\ge
2\prod_{i=1}^m |\sigma_{i,i+1}|.
\]
These extra constraints are described as hidden relations induced by the latent parametrization rather than by the explicit graph-imposed zeros [1001.3195].

In zeon algebra, graph-encoded nilpotent coefficients lead to another notion of hidden zero structure. For a monic zeon polynomial \(p(u)\) with scalar part \(f(u)=E(p(u))\), a simple zero \(\lambda_0\) of \(f\) lifts to a unique zeon zero
\[
\lambda=\lambda_0+A_1+\cdots+A_n,
\]
where the grade-\(k\) corrections satisfy
\[
A_k=-g(\lambda_0)^{-1}\bigl(p(\lambda_0+A_1+\cdots+A_{k-1})\bigr)_{(k-1)}.
\]
The scalar root is visible, while the nilpotent corrections are determined grade by grade by the graph-encoded zeon coefficients [2103.11227].

## 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 [2508.12894]. 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-\(z\) BCFW scaling [2604.01133]. At one loop in \(\mathrm{Tr}(\phi^3)\), hidden zeros are equivalent to unitarity for local surface integrands, and there is evidence that locality itself is already encoded by the zeros [2503.03805].

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 \(a^{(k_1,k_2)}_{jln}\). 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 \(n=2\) sector, so additional higher-point input is required for complete rigidity [2601.15088].

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.

Source: https://www.emergentmind.com/topics/graph-based-hidden-zeros