OPE Convergence in Conformal Field Theory

TL;DR

This paper proves that the operator product expansion and conformal block decomposition in unitary conformal field theories converge exponentially fast within a finite region, with explicit bounds depending on operator positions.

Contribution

It provides a rigorous explanation and explicit bounds for the convergence properties of OPE and conformal blocks in any unitary conformal field theory.

Findings

Convergence of OPE and conformal blocks is established in finite regions.

Convergence is exponentially fast with respect to operator dimension.

Explicit bounds depend on operator insertion positions.

Abstract

We clarify questions related to the convergence of the OPE and conformal block decomposition in unitary Conformal Field Theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above Delta contribute to correlation functions at most exp(-a Delta). Here the constant a>0 depends on the positions of operator insertions and we compute it explicitly.