Exploring Perturbative Conformal Field Theory in Mellin space

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.

arXiv comment47 pages, 15 figures

References in corpus (25)

• Writing CFT correlation functions as AdS scattering amplitudes
• Conformal Regge theory
• A Natural Language for AdS/CFT Correlators
• Spinning AdS Propagators
• Unitarity and the Holographic S-Matrix
• Further evidence for asymptotic safety of quantum gravity
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Leading proton production in deep inelastic scattering at HERA
• The S-matrix Bootstrap I: QFT in AdS
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• Analyticity and the Holographic S-Matrix
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• From correlation functions to event shapes
• Event shapes in N=4 super-Yang-Mills theory
• AdS Field Theory from Conformal Field Theory
• Energy-energy correlations in N=4 SYM
• Four point function of $\mathcal{N}=4$ stress-tensor multiplet at strong coupling
• On Feynman rules for Mellin amplitudes in AdS/CFT
• Mellin Amplitudes for Dual Conformal Integrals
• D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models
• Star Integrals, Convolutions and Simplices
• Unitarity and positivity constraints for CFT at large central charge
• Superstring Amplitudes as a Mellin Transform of Supergravity
• Lessons from crossing symmetry at large N
• Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature

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• Loops in AdS from Conformal Field Theory
• How to Succeed at Holographic Correlators Without Really Trying
• TASI lectures on AdS/CFT
• On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• A Note on (Non)-Locality in Holographic Higher Spin Theories
• Holographic Four-Point Functions in the (2, 0) Theory
• AdS${}_4$/CFT${}_3$ for Unprotected Operators
• Mellin Amplitudes for Fermionic Conformal Correlators
• Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS
• Constructing CFTs from AdS flows