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

Yang Li, Laurentiu Rodina

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.

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

Abstract

We show that cosmological wavefunctions in theories naturally generalize flat-space scattering amplitudes: via a simple map from tube variables to Mandelstam invariants, each wavefunction coefficient becomes an on-shell amplitude-like object associated with a generating graph . 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 zero at factorizes the generating graph, , and is equivalent to the shuffle decomposition . 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- 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.

Paper Structure

This paper contains 19 sections, 55 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Review: The cosmological wavefunction
  3. Kinematic Wavefunctions
  4. Kinematic wavefunctions from CHY
  5. Wavefunction blob zeros
  6. Dual factorization
  7. Dual factorization for wavefunctions
  8. Near-zero factorization from dual factorization
  9. Uniqueness from hidden zeros
  10. Unitarity from dual factorization
  11. Equivalence to BCFW scaling
  12. Outlook
  13. Appendix 1: The cosmological wavefunction
  14. Two maps from cosmology to amplitudes
  15. Appendix 2: Unification of cosmological zeros
  16. ...and 4 more sections

Figures (10)

  • Figure 1: All maximal tubings of a 222-star.
  • Figure 2: The 3-chain $\psi(123)$ is mapped to an ordered Tr($\phi^3$) 4-point amplitude $A[1234]$. Fixing vertex 1, the 2222-star is mapped to $A[15(2)(3)(4)6]$, which involves 6 different orderings, the permutations of $\{2,3,4\}$.
  • Figure 3: Left: A blob zero uniquely defined by $\partial A=\partial B=\{5,8\}$. Right: The choice $i,j=\{3,9\}$ splits the graph into the three branches of vertex 3; $A=\{1,2\}, B=\{4,5,6,7,8\}$ gives one of the zeros. The colored notation corresponds to (${\color{red} A},i,{\color{myblue} B},j$).
  • Figure 4: 5-point amplitude $D$-subsets for zero $p_1\cdot p_3=p_1\cdot p_4=0$. Dashed lines represent different choices of inserting leg 1. The colored notation corresponds to (${\color{red} A},i,{\color{myblue} B},j$). Each D-subset simply corresponds to the shuffle between two diagrams, the top one corresponding to $(i,A,j)$, the bottom one to $(i,B,j)$.
  • Figure 5: LHS: A $D$-subset from $\mathcal{B}_n$. RHS: The shuffle product of two lower-point diagrams, one from $\mathcal{B}_{(i,A,j)}$ and the other from $\mathcal{B}_{(i,B,j)}$, generating the subset on the LHS. The colored notation corresponds to (${\color{red} A},i,{\color{myblue} B},j$). $A_1\cup\dots \cup A_{\ell}=A$, and $B_1\cup \dots \cup B_r=B$.
  • ...and 5 more figures