New method for the conformal bootstrap with OPE truncations

TL;DR

This paper explores properties of 4-point correlators at the crossing symmetric point in conformal field theories and introduces a novel OPE truncation method for the conformal bootstrap, demonstrated on the 2D Ising model.

Contribution

It proposes a new OPE truncation approach for the conformal bootstrap based on error minimization, extending Gliozzi's method.

Findings

Analyzed 4-point correlators at the crossing symmetric point.

Developed an error-based OPE truncation technique.

Applied the method successfully to the 2D Ising CFT.

Abstract

We investigate two aspects of conformal field theories. In the first part, we study the general 4-point correlator of identical scalars around the fully crossing symmetric point $u=v=1$, where $u,v$ are conformally invariant cross ratios. Since this point is fully crossing invariant, we can deduce some general properties of the 4-point correlators from crossing symmetry. In the second part, we discuss the conformal bootstrap with OPE truncations. As a generalization of Gliozzi's method, we propose to extract the low-lying CFT data by minimizing the "error" induced by an OPE truncation. The error function $\eta$ measures the violation of crossing symmetry. The geometric interpretation of $\eta$ is the length of the vector associated with the truncated OPE. As an example, we apply the error-minimization method to the 2d Ising CFT with severely truncated OPEs.