OPE Convergence in Conformal Field Theory

TL;DR

This work proves that the OPE and conformal block decompositions in unitary CFTs converge in a finite region and do so exponentially fast in operator dimensions, by recasting correlators as overlaps in radial quantization on the cylinder. Through a spectral-density analysis and Hardy-Littlewood tauberian arguments, the authors bound the tails of the OPE and then of the conformal blocks, with an explicit dependence on insertion geometry via the mapping to a unit-disk variable $\rho(z)$. They show that high-dimension contributions are exponentially suppressed, justifying truncations in practical bootstrap computations and clarifying the behavior of weighted versus unweighted state densities. The results provide rigorous underpinnings for the bootstrap program and illuminate how high-dimension operators effectively decouple, with implications for effective CFTs and holographic dualities.

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.

Figures (6)

• Figure 1: Using conformal freedom, three operators can be fixed at $x_1=0$, $x_3=(1,0,\ldots,0)$, $x_4\to \infty$, while the fourth point $x_2$ can be assumed to lie in the (12) plane. The variable $z$ is then the complex coordinate of $x_2$ in this plane, while $\bar{z}$ is its complex conjugate.
• Figure 2: An $n$-point correlation function can be viewed as the overlap of two states in the radial quantization Hilbert space, living on a sphere.
• Figure 3: The map between $\mathbb{R}^D$ and the cylinder.
• Figure 4: Example of a reflection-positive six point function on the cylinder.
• Figure 5: Here we are considering four operators inserted at the shown points in a plane passing through the origin. We give complex coordinates in this plane. The circles are intersections of the spheres of radius 1 and $r$ with the plane. The angle $\alpha$ is arbitrary.