Hidden Zero Patterns in Scattering Amplitudes

• The paper demonstrates that hidden zero patterns arise from nontrivial algebraic cancellations and universal diagrammatic shuffles across scalar, NLSM, and YM amplitudes.
• It employs a unique shuffle factorization in Feynman diagrams, revealing precise cancellation conditions in color-ordered Tr(φ³) and effective field theories.
• These patterns unveil geometric insights via positive geometry and kinematic associahedra, enabling recursive amplitude constructions with enhanced UV scaling.

Hidden Zero Patterns in Scattering Amplitudes

A hidden zero in scattering amplitudes refers to the vanishing of a multi-leg on-shell amplitude on a specific locus in kinematic space that is not associated with any physical pole or propagator singularity. This phenomenon was first discovered in color-ordered scalar theories, such as Tr$(\phi^3)$, and was subsequently found to have a universal footprint in non-linear sigma models (NLSM), Yang–Mills (YM), and extends to unordered (gravitational, Galileon) amplitudes under certain conditions. Hidden zeros are characterized by nontrivial algebraic cancellation among Feynman diagrams, factorization properties reminiscent of standard pole-based unitarity, geometric aspects rooted in positive geometry, and powerful implications for amplitude construction and uniqueness.

1. Definition and Core Mechanism

Hidden zeros occur whenever all Mandelstam invariants $s_{ab} = (k_a + k_b)^2$ (and, where relevant, certain polarization contractions) between partitions of external legs ("A" and "B") are set to zero:

$$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$

(for scalars; with additional conditions on polarizations for spinning theories). In color-ordered amplitudes, $A_n(1,\ldots,n)$, these constraints correspond to setting to zero all non-planar invariants $c_{ij} = -2 p_i \cdot p_j$ inside a maximal rectangle (or "causal diamond") in the kinematic mesh of planar and non-planar invariants (Arkani-Hamed et al., 2023, Rodina, 2024, Zhou, 2024, Zhou, 26 Apr 2026).

No individual Feynman diagram or Parke–Taylor term vanishes on such a locus; instead, the total amplitude exhibits exact cancellation upon summing all diagrams—a nontrivial algebraic fact tracing to a unique diagrammatic backbone that can be analyzed via shuffle factorizations (Zhou, 26 Apr 2026, Zhou, 2024).

In NLSM and more generally in effective field theories with higher-point interactions, analogous loci exist, with numerators containing combinations of dot products forming specific contact term structures.

In YM amplitudes, hidden zeros occur when, in addition to the kinematic constraints above, one imposes that all cross-polarization contractions also vanish in the same index ranges: $\epsilon_a \cdot k_b = 0$, $\epsilon_b \cdot k_a = 0$, $\epsilon_a \cdot \epsilon_b = 0$ (Zhang, 2024, Zhang, 2024).

For ordered tree-level Tr$(\phi^3)$ amplitudes:

• Picking two nonadjacent legs $i$ and $s_{ab} = (k_a + k_b)^2$0 and partitioning as above, imposing $s_{ab} = (k_a + k_b)^2$1 for $s_{ab} = (k_a + k_b)^2$2, $s_{ab} = (k_a + k_b)^2$3, causes the amplitude to vanish identically—the quintessential "hidden zero" (Arkani-Hamed et al., 2023, Zhou, 2024, Zhou, 26 Apr 2026).

This phenomenon is purely an on-shell nullification: it is not related to the soft limits (Adler zero), nor to the appearance of a physical threshold.

2. Feynman Diagrammatics and Algebraic Structure

The mechanism behind hidden zeros relies on a universal diagrammatic shuffle factorization. For any tree-level color-ordered amplitude in Tr$s_{ab} = (k_a + k_b)^2$4, NLSM, or YM, every graph has a unique backbone propagator line connecting $s_{ab} = (k_a + k_b)^2$5 to $s_{ab} = (k_a + k_b)^2$6. The sum over all diagrams, when grouped by the shuffle of $s_{ab} = (k_a + k_b)^2$7 and $s_{ab} = (k_a + k_b)^2$8 subtrees attached to this line, factorizes the amplitude into a product of propagator chains for $s_{ab} = (k_a + k_b)^2$9 and $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$0, times the off-shell squared momentum $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$1:

$$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$2

where $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$3 for external on-shell $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$4 (Zhou, 26 Apr 2026, Zhou, 2024). Thus, the entire amplitude vanishes on this hidden-zero locus.

In NLSM and YM, the presence of contact term numerators (with specific sums over dot products or Lorentz contractions) is handled by appropriately extending the factorization, exploiting the cancellation of cross-contractions by the same shuffle logic (Zhou, 2024, Zhou, 26 Apr 2026). The reduction to $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$5 can be made purely algebraic, without recourse to the physical dimension or explicit embedding into orthogonal subspaces (Zhou, 2024).

The same algebraic cancellation is visible in the CHY representation of amplitudes: the vanishing occurs as the support of the scattering equations pinches multiple punctures together, and Parke–Taylor or Pfaffian structures collapse to zero (Zhang, 2024).

3. Geometric and Cosmological Interpretation

Hidden zeros acquire a geometric interpretation in terms of the positive geometry of moduli spaces and the kinematic associahedron. For color-ordered amplitudes, the hidden-zero locus corresponds to a codimension-one face or a codimension-two intersection in the associahedral realization, where the canonical form vanishes due to polytope degeneracy (De et al., 30 Mar 2025, Arkani-Hamed et al., 2023). Surfaceology and graph associahedra further clarify these loci as polytopal degenerations, corresponding to "flattening limits" or collisions of facets (De et al., 30 Mar 2025).

In the cosmological context, each stripped coefficient $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$6 of a graph $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$7’s wavefunction can be mapped to a flat-space amplitude $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$8 by a tube-to-Mandelstam substitution. The hidden zero for a subgraph splitting with interface $$k_a \cdot k_b = 0 \quad \forall a \in A,\, b \in B,$$9 is characterized by $A_n(1,\ldots,n)$0 for $A_n(1,\ldots,n)$1 in one block and $A_n(1,\ldots,n)$2 in the other, causing $A_n(1,\ldots,n)$3 to vanish. The cancellation is a direct combinatorial image of the graphical factorization of $A_n(1,\ldots,n)$4 into $A_n(1,\ldots,n)$5 and $A_n(1,\ldots,n)$6 (Li et al., 1 Apr 2026).

This shuffle factorization (dual to unitarity) is algebraically and geometrically equivalent to the vanishing; it can be iterated until the amplitude is uniquely fixed by locality and zeros, without explicit use of standard unitary factorization (Li et al., 1 Apr 2026).

4. Factorization Near Zeros and “Splittings”

Relaxing a single invariant in the hidden zero locus leads to a "splitting" structure: the amplitude factorizes, but not across a physical pole. For ordered Tr$A_n(1,\ldots,n)$7,

$A_n(1,\ldots,n)$8

with $A_n(1,\ldots,n)$9 planar propagators, and $c_{ij} = -2 p_i \cdot p_j$0 lower-point amplitudes on appropriate arguments (Arkani-Hamed et al., 2023, De et al., 30 Mar 2025, Zhou, 2024, Jones et al., 5 May 2025).

In general, this factorization does not correspond to any Feynman diagram or pole in the standard sense. For higher-codimension loci (relaxing more than one invariant), further splittings arise ("3-split", "smooth splitting") into three or more products, always with precise combinatorial and diagrammatic interpretation (Zhou, 2024, Jones et al., 5 May 2025).

In the context of the NLSM, analogous smooth splittings exist, though with restrictions to even-odd channels due to the structure of vertices (Jones et al., 5 May 2025).

The near-zero factorization also acquires a precise recursive statement: as one invariant is turned back on, a unique product of currents or lower-point amplitudes is produced, and this process (upon iteration) sharply constrains or even uniquely determines the amplitude (Li et al., 1 Apr 2026, Jones et al., 5 May 2025).

5. Analytical, Recursive, and Uniqueness Properties

Hidden zeros are tightly linked to recursion relations and the constructibility of amplitudes. For Tr$c_{ij} = -2 p_i \cdot p_j$1, NLSM, and related EFTs, imposing all hidden zero loci (together with locality, or with a minimal set of physical-pole factorizations) uniquely fixes the tree-level amplitude up to normalization (Rodina, 2024, Li et al., 1 Apr 2026, Li et al., 18 Aug 2025).

In NLSM, this principle enables a BCFW-like recursion entirely free from spurious boundary terms: standard BCFW fails due to poor $c_{ij} = -2 p_i \cdot p_j$2 scaling, but by shifting in such a way that the amplitude attains zeros at $c_{ij} = -2 p_i \cdot p_j$3 (the hidden zeros), the recursion closes exclusively on physical-pole residues (Li et al., 18 Aug 2025, Feng et al., 19 Apr 2025).

For loop-level amplitudes, hidden zeros survive (with additional constraints involving the loop momenta), and their imposition suffices—along with locality—to imply unitarity, and possibly also the reverse: unitarity and locality may emerge from hidden zero conditions alone (Backus et al., 5 Mar 2025, Zhou, 15 Apr 2026).

Moreover, hidden zeros correspond precisely (on homogeneous ansätze) to enhanced ultraviolet scaling under BCFW shifts (one power more than expected); this "subset-enhanced" scaling is equivalent to the presence of a hidden zero, offering another analytic handle (Rodina, 2024, Li et al., 1 Apr 2026).

6. Extensions and Universality

The hidden zero mechanism extends broadly:

• Spinning theories (YM, Gravity): In massless YM, hidden zeros arise if polarization contractions across the partition are also set to zero (e.g., $c_{ij} = -2 p_i \cdot p_j$4) (Zhang, 2024, Zhang, 2024). For gravity and unordered amplitudes, the structure persists when expanded (e.g., via double copy) onto a bi-adjoint scalar (BAS) basis; all hidden zeros of the constituent BAS amplitudes must vanish, and this property carries to the full gravitational amplitude typically after canceling spurious poles (Huang et al., 11 Feb 2025, Bartsch et al., 2024, Li et al., 2024).
• Effective field theories (DBI, special Galileon): The universal expansion of amplitudes onto the BAS basis, or via KLT/double copy, supplies a mechanism by which hidden zeros propagate to the composite theories they generate (Huang et al., 11 Feb 2025, Bartsch et al., 2024, Li et al., 2024).
• Massive Theories: Hidden zeros persist in Tr$c_{ij} = -2 p_i \cdot p_j$5, NLSM, and spontaneously broken gauge theories under symmetry-controlled mass deformations (e.g., uniform scalar mass, Kaluza–Klein reductions, or via spurion-induced potentials), whereas explicit Proca masses typically spoil them (González et al., 23 Jan 2026).

Their universality is further demonstrated by recursive and geometric arguments: the zeros, local propagator expansion, and (optionally) BCJ relations, together uniquely fix nearly all amplitude data in these models (Rodina, 2024, Zhang, 2024, Li et al., 1 Apr 2026).

7. Relations to Double Copy, Soft Theorems, and Algebraic Structures

Hidden zeros are intimately connected to the color–kinematics (BCJ) relations: the BCJ structure recursively propagates the vanishing to permutation-invariant theories, including gravity and special Galileon, via the Kawai–Lewellen–Tye (KLT) relations (Bartsch et al., 2024, Li et al., 2024).

The lowest-codimension hidden zero (single leg soft) coincides with the Adler zero in NLSM; this is shown to descend directly from the Weinberg soft graviton theorem via Lorentz-covariant transmutation operators (Cheung et al., 2017). Higher-codimension hidden zeros generalize Adler's result to "multi-soft" settings and factorization loci of the kinematic mesh ("causal diamonds") (Li et al., 2024).

Algebraically, the vanishing on these loci reflects the structure of the underlying cubic or quartic vertices, shuffle factorizations, and on-shell reducibility (currents, Berends–Giele expansions, block-shuffle identities) (Li et al., 2024, Zhou, 26 Apr 2026).

These structures are accessible both combinatorially (via diagrammatic grouping/shuffling) and via analytic methods (CHY, soft recursion, positive geometry). They impose nontrivial polynomial identities among amplitude building blocks and manifest themselves in the cancellation of residues on higher-codimension boundary strata of moduli space (Zhang, 2024, De et al., 30 Mar 2025).

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References:

• (Arkani-Hamed et al., 2023) Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons
• (Zhou, 2024) Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams
• (Zhou, 26 Apr 2026) Universal Interpretation of Hidden Zero and 2-Split of Tree-Level Amplitudes Using Feynman Diagrams, Part I
• (Rodina, 2024) Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes in Tr(φ³) theory
• (Li et al., 1 Apr 2026) Cosmological Wavefunctions as Amplitudes: Dual Shuffle Factorization and Uniqueness from New Hidden Zeros
• (Bartsch et al., 2024) Hidden Amplitude Zeros From Double Copy
• (Zhang, 2024, Zhang, 2024) New Factorizations of Yang-Mills Amplitudes and On the New Factorizations of Yang-Mills Amplitudes
• (Backus et al., 5 Mar 2025, Zhou, 15 Apr 2026) Emergence of Unitarity and Locality from Hidden Zeros at One-Loop; Towards New Hidden Zero and 2-Split of Loop-Level Feynman Integrands in Tr(φ³) Model
• (Jones et al., 5 May 2025, Li et al., 18 Aug 2025) Smooth Splitting and Zeros from On-Shell Recursion; A new recursion relation for tree-level NLSM amplitudes based on hidden zeros
• (Huang et al., 11 Feb 2025) Note on hidden zeros and expansions of tree-level amplitudes
• (González et al., 23 Jan 2026) Hidden Zeros in Massive Theories
• (Li et al., 2024) Diagrammatic Derivation of Hidden Zeros and Exact Factorisation of Pion Scattering Amplitudes
• (Donà et al., 2015) Scattering amplitudes in super-renormalizable gravity
• (Cheung et al., 2017) Unifying Relations for Scattering Amplitudes
• (Li et al., 2024) Hidden Zeros in Scaffolded General Relativity and Exceptional Field Theories
• (De et al., 30 Mar 2025) Hidden Zeros of the Cosmological Wavefunction