Feynman-like parameterizations of (anti-)de Sitter Witten diagrams for all masses at any loop order

Abstract

Computing correlation functions in curved spacetime is central to both theoretical and experimental efforts, from precision cosmology to quantum simulations of strongly coupled systems. In anti-de Sitter (AdS) and de Sitter (dS) space, the key observables, boundary correlators in AdS and late-time correlators in dS, are obtained via Witten diagram calculations. While formally analogous to flat-space Feynman diagrams, even tree-level Witten diagrams are significantly more complicated due to the structure of bulk propagators. Existing computational approaches often focus on scalars with a specific "conformal" mass, for which propagators simplify enough to permit the use of standard flat-space techniques. This restriction, however, omits the generic internal-line masses that arise in many cosmological and holographic settings. We present the Witten-Feynman (WF) parameterization, a general representation of scalar Witten diagrams in (A)dS as generalized Euler integrals. The WF framework applies in both position and momentum space, accommodates arbitrary internal and external masses, and holds at any loop order. It directly generalizes the familiar Feynman parameterization form of Feynman integrals, making it possible to import a broad range of amplitude techniques into the curved-space setting. We illustrate the method through two applications: a generalization of Weinberg's theorem on ultraviolet convergence and a series expansion technique that can yield explicit evaluations. Our results provide a unified computational tool for (A)dS boundary correlators, opening the door to more systematic calculations relevant for upcoming experiments and simulations.