The Unique Polyakov Blocks

TL;DR

The paper resolves the ambiguity in Polyakov bootstrap by introducing cyclic Polyakov blocks in Mellin space and provides a closed-form construction for arbitrary spin, together with a prescription to fix contact terms via cyclic exchanges and dispersion relations. It derives the conformal block decomposition of these blocks, computes direct-channel OPE data for double-twist operators, and clarifies analytic vs. non-analytic-in-spin contributions, including explicit results for scalar, vector, and spin-2 exchanges. It also reveals a deep link between cyclic Polyakov blocks and analytic bootstrap functionals, showing that these blocks act as generating functions for dual bootstrap functionals and enabling precise sum rules. The results pave the way for applying the Polyakov bootstrap to higher-dimensional CFTs and to investigations of Wilson-Fisher-type fixed points using the dual functional perspective.

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.

Figures (1)

• Figure 1: Plot of $\gamma_{0,\ell^\prime|\tau,0}=\gamma^{\text{anal.}}_{0,\ell^\prime|\tau,1}+\gamma^{\text{n.-a.}}_{0,\ell^\prime|\tau,0}$ for $d=3$, $\Delta=2$ and varying $\tau$ along the $x$ axis. In blue we have $\ell^\prime=0$, in orange $\ell^\prime=1$ and in green $\ell^\prime=2$. We see that for $\ell>0$, where there are only the analytic in spin contributions \ref{['scalar_anal']}, $\gamma_{0,\ell^\prime|\tau,0}$ is always positive and has double zeros at double-twist values of $\tau$. For $\ell^\prime=0$ there is also a non-analytic contribution \ref{['gammaEll']}, so that the full anomalous dimension $\gamma_{0,\ell^\prime|\tau,0}$ also has single zeros at some double-twist values of $\tau$.