Handbook of derivative AdS amplitudes

TL;DR

This paper extends the computation of Witten diagrams in AdS space to include bulk interactions with derivatives, providing explicit formulas and a Mathematica package, with implications for holographic cosmology.

Contribution

It introduces new explicit formulas for derivative-involving Witten diagrams in AdS and discusses their renormalization, expanding previous scalar-only analyses.

Findings

Closed-form expressions for certain AdS amplitudes in 3D

A Mathematica package for all derived amplitudes

Finite correlators may still require renormalization

Abstract

In the 2022 study, together with Paul McFadden and Kostas Skenderis, I analyzed tree-level 3- and 4-point Witten diagrams (amplitudes) of scalar operators in anti-de Sitter space in momentum space. This paper constitutes its extension to Witten diagrams with bulk interactions involving spacetime derivatives. In $d = 3$ boundary dimensions the Witten diagrams involving conformally coupled and massless scalars can be evaluated in closed form. Such cases are of interest in holographic cosmology and correspond to dual operators of conformal dimensions $\Delta = 2$ and $3$ respectively. I present explicit formulae for all such amplitudes and provide a Mathematica package serving as the repository of all the results. I discuss renormalization issues and show that, contrary to the expectation, even finite correlators may acquire non-trivial renormalization effects.