Conformal Four Point Functions and the Operator Product Expansion

TL;DR

This paper analyzes four-point functions in conformally invariant theories, deriving recurrence relations and explicit solutions in 2D and 4D, and applies these to free fields and AdS/CFT results.

Contribution

It introduces a recurrence relation for conformal four-point functions involving arbitrary spin fields and provides explicit solutions in specific dimensions.

Findings

Explicit hypergeometric function solutions in 2D and 4D

Recurrence relations for operator contributions

Analysis of AdS/CFT four-point functions

Abstract

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.