Multivariate hypergeometric solutions of cosmological (dS) correlators by $\text{d} \log$-form differential equations

TL;DR

This paper derives analytic solutions for cosmological correlators using multivariate hypergeometric functions, employing $ ext{d} ext log$-form differential equations and novel techniques like power series expansion and blow-up methods.

Contribution

It provides a systematic approach to solve arbitrary tree-level cosmological correlators with massive propagators using multivariate hypergeometric functions and introduces new techniques for handling complex differential equations.

Findings

Analytic solutions expressed as multivariate hypergeometric functions.

Factorization property of solutions for multiple vertices.

Development of methods for solving $ ext{d} ext log$-form differential equations.

Abstract

In this paper, we give the analytic expression for the homogeneous part of solutions of arbitrary tree-level cosmological correlators, including massive propagators and time-derivative interaction cases. The solutions are given in the form of multivariate hypergeometric functions. It is achieved by two steps. Firstly, we indicate the factorization of the homogeneous part of solutions, i.e., the homogeneous part of solutions of multiple vertices is the product of the solutions of the single vertex. Secondly, we give the solution to the $\text{d} \log$-form differential equations of arbitrary single vertex integral family. We also show how to determine the boundary conditions for the differential equations. There are two techniques we developed for the computation. Firstly, we analytically solve $\text{d} \log$-form differential equations via power series expansion. Secondly, we handle degenerate multivariate poles in power series expansion of differential equations by blow-up. They could also be useful in the evaluation of multi-loop Feynman integrals in flat spacetime.