The Unique Polyakov Blocks

TL;DR

This paper derives a closed-form expression for Polyakov blocks in Mellin space for any spin and scaling dimensions, providing methods to resolve contact term ambiguities and connecting to dispersion relations and bootstrap functionals.

Contribution

It introduces a novel prescription for fixing contact term ambiguities in Polyakov blocks and relates cyclic amplitudes to dispersion relations, advancing the analytic bootstrap approach.

Findings

Explicit Mellin space expression for Polyakov blocks of arbitrary spin.

A new method to fix contact term ambiguities uniquely.

Extension of OPE data extraction to non-analytic in spin contributions.

Abstract

In this work we present a closed form expression for Polyakov blocks in Mellin space for arbitrary spin and scaling dimensions. We provide a prescription to fix the contact term ambiguity uniquely by reducing the problem to that of fixing the contact term ambiguity at the level of cyclic exchange amplitudes -- defining cyclic Polyakov blocks -- in terms of which any fully crossing symmetric correlator can be decomposed. We also give another, equivalent, prescription which does not rely on a decomposition into cyclic amplitudes and we underline the relation between cyclic amplitudes and dispersion relations in Mellin space. We extract the OPE data of double-twist operators in the direct channel expansion of the cyclic Polyakov blocks using and extending the analysis of \cite{Sleight:2018epi,Sleight:2018ryu} to include contributions that are non-analytic in spin. The relation between cyclic Polyakov blocks and analytic Bootstrap functionals is underlined.