Cosmological Scattering Equations at Tree-level and One-loop

TL;DR

The paper extends CHY-inspired worldsheet methods to cosmological correlators in de Sitter space, formulating a tree-level worldsheet approach for massive $\phi^4$ theory via cosmological scattering equations (CSE) and an operatorial Pfaffian. A double-cover reformulation yields efficient graphical rules that reproduce Witten-diagram expansions up to eight points and clarifies factorization structures, including ladder and non-ladder topologies. A 1-loop generalization is proposed using a split representation and auxiliary punctures, with verification at four points and validity for $0\le m\le d/2$ in dS (BF bound after Wick rotation to AdS). The framework connects cosmological correlators to flat-space CHY techniques, offering computational advantages and opportunities to generalize to spinning theories and broader mass spectrums.

Abstract

We recently proposed a formula for tree-level $n$-point correlators of massive $φ^4$ theory in de Sitter momentum space which consists of an integral over $n$ punctures on the Riemann sphere and differential operators in the future boundary dubbed the cosmological scattering equations. This formula was explicitly checked up to six points via a map to Witten diagrams using the global residue theorem. In this work we provide further details of these calculations and present an alternative formulation based on a double cover of the Riemann sphere. This framework can be used to derive simple graphical rules for evaluating the integrals more efficiently. Using these rules, we check the validity of our formula up to eight points and sketch the derivation of $n$-point correlators. Finally, we propose a similar formula for 1-loop $n$-point correlators in terms of an integral over $(n+2)$ punctures on the Riemann sphere, which we verify at four points. The 1-loop formula holds for small masses in de Sitter space and arbitrary masses satisfying the Breitenlohner-Freedman bound after Wick-rotating to Anti-de Sitter space.

Figures (25)

• Figure 1: Graph representation of ${\cal A}(\mathbb{I}_{n}:1\,p-2,2\,p+2,3\,n-1,\ldots,n\,p)$.
• Figure 2: Graph representation of ${\cal A}(\mathbb{I}_n: 1p-2, 2p+2, 3 n-1,\ldots, np)$ in the double cover language.
• Figure 3: This particular factorization cut vanishes trivially since there is only one fixed puncture on the upper side of the dashed black line (rule I).
• Figure 4: This factorization cut vanishes since the dashed black line cuts six lines (more than four), five black and one red (rule II).
• Figure 5: Factorization of ${\cal A}(\mathbb{I}_4)$ with (a) ${\rm Pf} A^{14}_{14}$, (b) ${\rm Pf} A^{12}_{12}$.