Exploring Perturbative Conformal Field Theory in Mellin space

TL;DR

This paper develops a comprehensive framework for representing correlation functions in conformal field theories using Mellin space, providing Feynman rules, and exploring non-conformal cases with scalar perturbations.

Contribution

It introduces a complete set of Feynman rules for Mellin amplitudes in scalar conformal field theories at tree level and examines non-conformal extensions.

Findings

Derived factorised Mellin amplitudes with beta functions for scalar propagators.

Proved Feynman rules for tree-level scalar correlation functions in Mellin space.

Extended analysis to non-conformal perturbations, involving hypergeometric functions.

Abstract

We explore the Mellin representation of correlation functions in conformal field theories in the weak coupling regime. We provide a complete proof for a set of Feynman rules to write the Mellin amplitude for a general tree level Feynman diagram involving only scalar operators. We find a factorised form involving beta functions associated to the propagators, similar to tree level Feynman rules in momentum space for ordinary QFTs. We also briefly consider the case where a generic scalar perturbation of the free CFT breaks conformal invariance. Mellin space still has some utility and one can consider non-conformal Mellin representations. In this context, we find that the beta function corresponding to conformal propagator uplifts to a hypergeometric function.