Wilson network expansion for four-point contact and exchange scalar Feynman diagrams in AdS$_2$

TL;DR

This paper develops a Wilson network expansion method for four-point scalar Feynman diagrams in AdS$_2$, providing new integral identities and series expansions that connect to conformal blocks.

Contribution

It introduces a novel Wilson network expansion for AdS$_2$ Feynman diagrams, linking them to matrix elements of Wilson line networks with conformal weights.

Findings

Derived new integral identities for AdS propagators.

Expanded four-point diagrams into series of Wilson line network operators.

Reproduced conformal block decompositions of Witten diagrams.

Abstract

We derive new integral identities for AdS propagators and further develop the Wilson network expansion for AdS Feynman diagrams. In particular, we demonstrate that four-point contact and exchange scalar diagrams in two dimensions can be expanded into several infinite series of matrix elements of Wilson line network operators with running conformal weights. Each series is characterized by specific multi-trace operators associated with the external and intermediate edges of the corresponding graphs. The resulting expansions near the conformal boundary reproduce the well-known decompositions of the corresponding four-point Witten diagrams into conformal blocks.