Notes On de-Sitter Mellin Barnes Amplitudes

TL;DR

This paper develops a Mellin space method for boundary correlators in de Sitter and anti-de Sitter spaces, revealing their analytic structure and enabling computation of scalar correlators via Mellin-Barnes amplitudes, bridging AdS and dS results.

Contribution

It introduces a Mellin space framework for dS boundary correlators, connecting AdS and dS via analytic continuation, and applies the split representation for Witten diagram evaluations.

Findings

Derived Mellin-Barnes amplitudes for dS correlators.

Established analytic continuation phases between AdS and dS.

Facilitated computation of scalar three- and four-point functions.

Abstract

In this paper, we create a Mellin space method for boundary correlation functions in de Sitter (dS) and anti-de Sitter (AdS) spaces. We demonstrate that the analytic continuation between AdS${}_{d+1}$ and dS${}_{d+1}$ is encoded in a set of simple relative phases using the Mellin-Barnes representation of correlators. It helps us to determine the scalar three-point and four-point functions and their corresponding Mellin-Barnes amplitudes in dS${}_{d+1}$ space using the known results from AdS${}_{d+1}$ space. The Mellin-Barnes representation reveals the analytic structure of boundary correlation functions over all $d$ and scaling dimensions. In the present discussion, the {\it split representation} have been used as an instrumental technique in particularly the evaluation of bulk Witten diagrams and is suitable to obtain the {\it Conformal Partial Wave decomposition} of tree-level exchange in the bulk Witten diagrams. The equivalent adjustment to the cosmological three-point and four-point function of generic external scalars may be further extracted from these results, assuming the weak breakdown of the de Sitter isometries. These findings offer a step towards a more methodical comprehension of de Sitter observables utilising Mellin space techniques at the tree level and beyond.