Cosmological Scattering Equations at Tree-level and One-loop

TL;DR

This paper extends the cosmological scattering equations framework to include one-loop correlators in de Sitter space, providing new formulas and graphical evaluation rules validated up to eight points.

Contribution

It introduces a novel formula for one-loop n-point correlators in de Sitter space using scattering equations, expanding the tree-level approach.

Findings

Validated the one-loop formula at four points.

Developed graphical rules for efficient integral evaluation.

Extended the framework to higher points up to eight.

Abstract

We recently proposed a formula for tree-level $n$-point correlators of massive $\phi^4$ 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.