Aspects of the conformal operator product expansion in AdS/CFT correspondence

TL;DR

Using the AdS/CFT framework, the paper analyzes a scalar conformal four-point function by evaluating AdS exchange graphs and performing conformal partial wave analysis. It shows that AdS exchange graphs are analytic in both direct and crossed channels, enabling a consistent OPE on the boundary and linking logarithms to anomalous dimensions. The authors introduce a general method to compute scalar and tensor exchange contributions to CPWA and derive a triangular system to determine couplings and anomalous dimensions, concluding that bulk KK modes appear non-renormalized in the boundary OPE. The work suggests extensions to AdS5/CFT4 and AdS4/CFT3 and discusses shadow contributions, with an appendix establishing cancellations in crossed channels via hypergeometric identities.

Abstract

We present a detailed analysis of a scalar conformal four-point function obtained from AdS/CFT correspondence. We study the scalar exchange graphs in AdS and discuss their analytic properties. Using methods of conformal partial wave analysis, we present a general procedure to study conformal four-point functions in terms of exchanges of scalar and tensor fields. The logarithmic terms in the four-point functions are connected to the anomalous dimensions of the exchanged fields. Comparison of the results from AdS graphs with the conformal partial wave analysis, suggests a possible general form for the operator product expansion of scalar fields in the boundary CFT.

Figures (2)

• Figure 1: The $A$ and $B$ graphs. In the $A$ graph the solid lines correspond to the full ${\cal O}_{\phi}(x)$ propagator (\ref{['2ptphi']}). In the $B$ graph, solid lines correspond to the "bulk-to-boundary" propagators (\ref{['btbo']}) and the dotted line to the "bulk-to-bulk" one (\ref{['btbu']})
• Figure 2: The tensor exchange graphs. The dark blobs correspond to the full vertex functions obtained by suitable amputation of (\ref{['mvertex']}).