Exploring Perturbative Conformal Field Theory in Mellin space

TL;DR

This work develops a perturbative framework for conformal field theories in Mellin space, proving that tree-level amplitudes for scalar operators factorize into a product of beta-function propagators linked to internal lines. It provides a diagrammatic, Schwinger-parameter algorithm to write Mellin amplitudes for arbitrary tree diagrams and clarifies the dual Mellin-momentum interpretation, including a clean spectral interpretation of exchanged primaries and descendants. The authors also extend the formalism to non-conformal settings, where propagator beta-functions uplift to hypergeometric functions and scale-invariant (off-shell) amplitudes exhibit LSZ-like structure. At one loop, they derive an integral representation and perform consistency checks, with a special star-delta case yielding exact results. Overall, the paper lays a foundation for Mellin-space perturbation theory in CFTs and points toward extensions to tensors, loops, and strong-coupling connections.

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.

Figures (16)

• Figure 1: Contact Interaction Diagram
• Figure 2: Two vertex
• Figure 3: Three vertex tree
• Figure 4: Skeleton of the single propagator diagram
• Figure 5: Tree level three vertex