Differential Equations for Massive Correlators

TL;DR

This paper reveals a combinatorial and graphical framework for deriving and solving differential equations governing massive scalar field correlators in de Sitter space, simplifying their analysis.

Contribution

It introduces a novel combinatorial and graphical approach to derive and solve differential equations for massive cosmological correlators, enhancing computational efficiency.

Findings

Derived a finite set of master integrals obeying first-order differential equations.

Presented a graphical description using graph tubings to encode couplings and singularities.

Solved the system explicitly in small and large mass limits.

Abstract

We uncover a combinatorial structure governing the differential equations satisfied by wavefunction coefficients of scalar fields with generic masses in de Sitter space. Using an integral representation of the massive mode functions, we express the Feynman integrals underlying cosmological correlators as twisted integrals of rational functions. In this formulation, the integrals belong to a finite set of master integrals obeying a first-order system of differential equations, which can be derived efficiently in the time-integral representation. We show that these equations admit a simple graphical description in terms of graph tubings, which encode the couplings among basis functions and the evolution of singularities. This structure provides an efficient algorithm to derive the differential equations, and a boundary-centric perspective on massive cosmological correlators in which their analytic structure emerges from underlying combinatorial data. As an illustration, we solve the system in the limits of small and large masses.