Crossing Symmetric Dispersion Relations for Mellin Amplitudes

TL;DR

This paper develops crossing symmetric dispersion relations for Mellin amplitudes in conformal field theories, enabling a non-perturbative bootstrap approach that clarifies contact term ambiguities and connects to AdS Witten diagrams.

Contribution

It introduces a new crossing symmetric dispersion relation framework for Mellin amplitudes, fixing contact ambiguities and linking Polyakov blocks to AdS Witten diagrams.

Findings

Sum rules from new and traditional dispersion relations are equivalent.

Framework provides bounds for Wilson coefficients in AdS effective field theories.

Connects Polyakov blocks with Witten diagrams in AdS space.

Abstract

We consider manifestly crossing symmetric dispersion relations for Mellin amplitudes of scalar four point correlators in conformal field theories (CFTs). This allows us to set up the non-perturbative Polyakov bootstrap for CFTs in Mellin space on a firm foundation, thereby fixing the contact term ambiguities in the crossing symmetric blocks. Our new approach employs certain "locality" constraints replacing the requirement of crossing symmetry in the usual fixed-$t$ dispersion relation. Using these constraints we show that the sum rules based on the two channel dispersion relations and the present dispersion relations are identical. Our framework allows us to connect with the conceptually rich picture of the Polyakov blocks being Witten diagrams in anti-de Sitter (AdS) space. We also give two sided bounds for Wilson coefficients for effective field theories in AdS space.