Remarks on the Convergence Properties of the Conformal Block Expansion

TL;DR

The paper refines convergence estimates for the conformal block expansion by isolating a coefficient-positivity structure in 3d conformal blocks. It derives an explicit closed-form for 3d blocks on the real axis via a fourth-order Casimir–second-order Casimir ODE, enabling a positive-coefficient representation and a tighter tail bound. By removing contributions from light scalars and factoring out a power of (1-r^2), it achieves a stronger Δ_* dependence in the convergence estimate, with dimension-dependent optimal choices for the factoring parameter γ. The results enhance the reliability of truncation-based bootstrap methods, especially in 3d, and provide a framework for similar improvements in other dimensions.

Abstract

We show how to refine conformal block expansion convergence estimates from hep-th/1208.6449. In doing so we find a novel explicit formula for the 3d conformal blocks on the real axis.