Perturbative Conformal Field Theory in the Mellin Space

Abstract

The aim of this project is to study the Mellin representation of correlation functions in perturbative scalar conformal field theory and investigate the existence of Feynman rules that can be associated with perturbative diagrams. It is known that the Mellin representation of correlation functions in CFTs makes the covariance with all conformal symmetries manifest. The constraints on the Mellin variables that make the covariance of the amplitude with special conformal symmetry manifest, can be interpreted as the conservation of ’Mellin momentum’. The poles of a propagator in the complex Mellin momentum plane correspond to the exchanged primary operator and its descendants. Thus the Mellin space furnishes a spectral representation of n-point functions in conformal field theories. In this project, we have been able to derive the Mellin amplitude for an arbitrary tree level diagram and have found that we can associate a set of Feynman rules to these diagrams. We have been investigating the existence of similar rules for one-loop diagrams. Preliminary investigations indicate that the Mellin amplitude for such diagrams can be expressed as Mellin Barnes integrals analogous to loop integrals in momentum space. However, we have not been able to establish these results yet. We expect that this formulation of perturbative CFT in the Mellin space can be employed in deriving a natural dual wordsheet description of string theory in Anti de Sitter space from a conformal field theory on its boundary in the large N limit.