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.
Highlights
It has been realised in the last few years, beginning with the pioneering work of Mack [1, 2], that Mellin space provides the natural setting for the study of Conformal Field theories (CFTs)
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 develop a diagrammatic algorithm to write down the Mellin amplitude for any Feynman diagram as an integral over Schwinger parameters corresponding to the internal propagators in the diagram
Summary
It has been realised in the last few years, beginning with the pioneering work of Mack [1, 2] (see [3]), that Mellin space provides the natural setting for the study of Conformal Field theories (CFTs). Following Mack, the application of the Mellin representation of conformal correlation functions was explored at strong coupling for large N CFTs using tree level Witten diagrams in AdS [4,5,6,7,8,9,10]. We present a complete derivation of the Feynman rules associated to tree level amplitudes in complete generality For this purpose, we develop a diagrammatic algorithm to write down the Mellin amplitude for any Feynman diagram (upto arbitrary loop order) as an integral over Schwinger parameters corresponding to the internal propagators in the diagram. We consider a tree level diagram with a single internal line (involving scalar fields) in Mellin space for such theories. More details on the notations and convention can be found in the appendix A
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