Cosmological Wavefunctions as Amplitudes: Dual Shuffle Factorization and Uniqueness from New Hidden Zeros

Abstract: We show that cosmological wavefunctions in $φ^n$ theories naturally generalize flat-space $\mathrm{Tr}(φ^3)$ scattering amplitudes: via a simple map from tube variables to Mandelstam invariants, each wavefunction coefficient $ψ{\mathcal{G}}$ becomes an on-shell amplitude-like object $\mathcal{A}_G$ associated with a generating graph $G$. At tree level these objects coincide with the Cachazo-He-Yuan construction based on Cayley functions that generalizes Parke-Taylor factors. We uncover new graph-based hidden zeros that extend and unify all known cosmological zeros. Based on this zero structure, we uncover a factorization principle dual to unitarity. Instead of factorization across poles, $A\to A_L\times A_R$, a zero at $p{a\in G_L}!\cdot! p_{b\in G_R}=0$ factorizes the generating graph, $G\to G_L\times G_R$, and is equivalent to the shuffle decomposition $\mathcal{A}G=\mathcal{A}{G_L}\unicode{x29E2}\mathcal{A}_{G_R}$. Near-zero factorization is a simple consequence of this new structure. Using dual factorization, we show that locality together with the full set of hidden zeros uniquely fixes tree-level cosmological wavefunctions without assuming unitarity. We show that these zeros are equivalent to special enhanced large-$z$ behavior under Britto-Cachazo-Feng-Witten (BCFW) shifts, extending the zeros--BCFW correspondence beyond flat-space amplitudes. We also find evidence for further extensions of the zero structure and loop-level uniqueness. Our results show that cosmology provides a natural arena for on-shell methods and even reveals new structure in flat-space amplitudes.

• The paper introduces the innovative concept that cosmological wavefunction coefficients uniquely map to amplitude-like objects using combinatorial generating graphs and blob zeros.
• It details how the shuffle product factorizes these amplitudes by leveraging tube variables and on-shell kinematics, replicating BCFW shift behavior.
• The study validates the method with numerical examples, showing that a minimal set of hidden zeros fully determines the structure of the wavefunction.

Cosmological Wavefunctions, Shuffle Dual Factorization, and Hidden Zero Uniqueness

Introduction

The paper "Cosmological Wavefunctions as Amplitudes: Dual Shuffle Factorization and Uniqueness from New Hidden Zeros" (2604.01133) formulates a comprehensive connection between the combinatorics of late-time cosmological wavefunctions and the analytic, graph-theoretic structure of scattering amplitudes. Leveraging the dual language of tube variables and on-shell kinematics, the authors show that cosmological wavefunction coefficients in scalar $\phi^n$ theories map to amplitude-like objects constructed from generating graphs. This enables a transfer of amplitude-theoretic principles—particularly the structure of hidden zeros, shuffle decompositions, and BCFW behavior—to the cosmological setting, and reveals a mechanism for uniquely determining wavefunctions from purely combinatorial and locality properties, without recourse to traditional diagrammatic unitarity.

Cosmological Wavefunctions: Combinatorial Structure

The main object of interest is the late-time vacuum wavefunction coefficient $\psi_G$, associated to a graph $G$ (i.e., a Feynman diagram stripped of universal 1-tube and total-energy singularities). The coefficients admit an explicitly combinatorial construction in terms of sums over maximal tubings of $G$, each defined by a product over tube energy variables $S_\tau$ for connected subgraphs $\tau$.

Figure 1: All maximal tubings of a 222-star.

For the archetype 222-star, this construction yields all possible maximal tubings, demonstrating the systematic reduction of diagrammatic complexity to purely combinatorial sums dictated by graph topology.

Kinematic Mapping: Amplitudes from Tube Variables

A central element is the introduction of a map from tube variables $S_I$ to flat-space Mandelstam invariants $s_I = (\sum_{i \in I} p_i)^2$, thus interpreting each stripped wavefunction as an amplitude-like quantity $\mathcal{A}_G$. The augmentation of $G$ by an auxiliary vertex implements on-shell conditions, facilitating direct correspondence with the Cayley function—realized as the Parke-Taylor factor for chains and its CHY generalization for arbitrary trees.

Figure 2: The 3-chain $\psi_G$0 maps to a color-ordered Tr($\psi_G$1) amplitude $\psi_G$2; for the 2222-star, multiple permutations are generated by the mapping.

This structural mapping demonstrates that cosmological wavefunctions are not just analogous but strictly equivalent, at the tree level, to unordered sums over color-ordered amplitudes generated by the underlying graph topology.

Hidden Zeros and Graph-Based Blob Zeros

The authors systematically generalize the concept of "hidden zeros"—kinematic loci where amplitude objects vanish nontrivially—to cosmological wavefunctions, defining blob zeros associated with separations of the underlying graph. For a pair of vertices $\psi_G$3 splitting $\psi_G$4 into subgraphs $\psi_G$5, $\psi_G$6, the condition $\psi_G$7 yields strong vanishing constraints. The construction encompasses all previously known zero types (parametric, wavefunction, and factorization zeros) and introduces new constraints critical for establishing uniqueness.

Figure 3: Left: Example of a blob zero uniquely defined by external boundary $\psi_G$8; Right: Multi-branching example splitting at vertices $\psi_G$9.

Dual Shuffle Factorization: The Principle Beyond Unitarity

Imposing graph-based zeros, the ansatz for $G$0 is shown to necessarily decompose as a shuffle product of objects associated with the factor graphs induced by the cut. Formally, the presence of a zero for $G$1 implies $G$2, where the shuffle product interleaves diagrams while preserving the internal ordering of each factor.

Figure 4: 5-point amplitude $G$3-subsets for the zero $G$4: different insertions generate shuffle products.

The duality is made precise: while traditional unitarity factorization occurs at poles, this dual factorization occurs at zeros and governs the diagrammatic generation (rather than only residues) of the full amplitude-like object from its subgraph constituents.

Figure 5: Dual factorization at the graph level: cutting a graph at specified vertices, the amplitude is reconstructed as the shuffle sum of lower-point diagrams.

Uniqueness from Zeros and Graph Reductions

Iteratively imposing all blob zeros for a given graph enforces a recursive decomposition into 2-vertex chain graphs, each of which is fixed up to normalization. The process provably fixes all coefficients of the local ansatz, as the number of independent coefficients reduces multiplicatively at each factorization.

Strong numerical results include the identification that, for an 11-vertex 222-2222-222 triple-star graph with 82,320 initial coefficients, only 8 zero conditions suffice for a complete fix.

Figure 6: Systematic reduction of the 5-chain via imposing blob zeros, demonstrating exact recursive factorization.

Figure 7: Reduction of the 234-star via sequential blob zeros, culminating in a unique assignment.

Figure 8: Decomposition of the 222-2222-222 triple-star graph under blob zeros—starting from 82320 coefficients, 8 zeros complete the fix.

Equivalence to BCFW Shift Behavior

Blob zeros exhibit direct correspondence with enhanced scaling under large-Britto-Cachazo-Feng-Witten (BCFW) shifts, extending the scaling-zeroes equivalence from amplitudes to cosmological wavefunctions. In particular, for graphs split into $G$5 branches at $G$6, the enhanced scaling is by $G$7 powers, tightly encoding all blob zero constraints into shift behavior.

Implications and Directions for Future Research

The results position zeros and dual shuffle factorization as organizing, constraining principles coequal in status to locality and unitarity—but acting through a dual channel. The explicit combinatorial-analytic correspondence enables recursive construction of unique cosmological wavefunctions at tree level and suggests a generic framework for higher loops. The relation with the CHY/Cayley formalism hints at extensive interplay between amplitude geometry, cosmological polytopes, and moduli space interpretations. The approach admits generalization to non-scalar theories, with possible implications for spin, color, and gravity. Furthermore, the observed loop-level phenomena warrant systematic exploration of the principle beyond tree order.

Conclusion

The paper demonstrates that the cosmological wavefunction in scalar theories is uniquely determined by locality and a minimal set of graph-based blob zeros, with dual shuffle factorization as the central organizing constraint. The constructive mapping from combinatorics to amplitudes, the unification and extension of hidden zeros, and the equivalence to on-shell BCFW behavior collectively advance the analytic understanding of both cosmological and flat-space observables. The structure discovered here is likely to inform future approaches to S-matrix bootstrap, positive geometry, and the combinatorial foundations of perturbative quantum field theory.