Moments in the CFT Landscape

TL;DR

This paper introduces a new numerical bootstrap method using moment observables to analyze conformal field theories, revealing novel features and spectral reorganizations in the operator spectrum across various dimensions.

Contribution

The paper develops a moment-based bootstrap framework that provides global bounds and uncovers new spectral phenomena in CFTs, extending beyond traditional methods.

Findings

Bounds on moments are robust in the heavy correlator regime.

Low-lying moments capture corrections to heavy limit results.

Discovery of persistent kinks indicating spectral reorganizations.

Abstract

We develop a novel numerical bootstrap for unitary, crossing-symmetric conformal field theories, focusing on moment observables defined as weighted averages over conformal data. Providing a global and coarse-grained probe of the operator spectrum, this framework yields numerically rigorous bounds on the operator distribution using standard semidefinite programming techniques. In the heavy correlator regime, these bounds remain robust and converge rapidly towards analytically-derived power laws. At finite external dimensions, low-lying moments capture corrections to analytic heavy limit results, while reproducing familiar bootstrap solutions such as Ising-model kinks on the boundary of moment space. Most importantly, the moment bootstrap reveals new features in previously unexplored regions of the bootstrap landscape. The lower bounds on moment variables exhibit two continuous families of kinks persisting across $2 < d < 6$, reflecting nontrivial spectral reorganizations connected to underlying operator decoupling phenomena. These results demonstrate that moment variables uncover bootstrap solutions and collective structures that are difficult to access within traditional numerical approaches.