Higher-Trace Operators and Cut Diagrammatics in the Conformal Block Expansion

We study four-point functions of identical scalar operators in conformal field theories with AdS duals in large-$N$ expansion. We analyze the appearance of higher-trace operators in theories dual to bulk ${\phi }^3$ and ${\phi }^4$ interactions, focusing on how these operators are required by crossing symmetry. We compute part of the OPE data associated with these operators. We also introduce a diagrammatic framework for organizing the different terms in the conformal block expansion within the large-$N$ expansion. This framework refines the use of crossing symmetry by allowing it to be applied to individual diagrammatic topologies, rather than only to the full correlator. It further separates different contributions to the OPE data by associating them with different cut diagrams. In this language, the emergence of higher-trace operators and their relation to lower-trace OPE data become more manifest.