On the Polyakov-Mellin bootstrap

TL;DR

This paper advances the Polyakov-Mellin bootstrap method in conformal field theory by analyzing the role of contact Witten diagrams and deriving explicit Mellin space expressions for crossing kernels, enhancing the understanding of holographic bootstrap.

Contribution

It demonstrates the necessity of contact Witten diagrams in Mellin space bootstrap and provides explicit hypergeometric function formulas for crossing kernels in this framework.

Findings

Contact Witten diagrams are essential in large c and epsilon expansions.

Derived explicit Mellin space expressions for crossing kernels.

Simplified Witten diagram representations facilitate bootstrap calculations.

Abstract

We elaborate on some general aspects of the crossing symmetric approach of Polyakov to the conformal bootstrap, as recently formulated in Mellin space. This approach uses, as building blocks, Witten diagrams in AdS. We show the necessity for having contact Witten diagrams, in addition to the exchange ones, in two different contexts: a) the large $c$ expansion of the holographic bootstrap b) in the $\epsilon$ expansion at subleading orders to the ones studied already. In doing so, we use alternate simplified representations of the Witten diagrams in Mellin space. This enables us to also obtain compact, explicit expressions (in terms of a ${}_7F_6$ hypergeometric function!) for the analogue of the crossing kernel for Witten diagrams i.e., the decomposition into $s$-channel partial waves of crossed channel exchange diagrams.