Diagonal Limit for Conformal Blocks in d Dimensions

TL;DR

The paper analyzes conformal blocks in d dimensions along the diagonal z = zbar, deriving ordinary differential equations that these diagonal blocks satisfy. The general case yields a fourth-order diagonal ODE, with a third-order equation for spin-0, enabling a complete on-diagonal description and efficient numerical evaluation. For the special case a = 0, the authors obtain closed-form diagonal blocks in terms of finite sums of {}_3F_2 functions and develop a recursion framework to extend to higher spins. They also propose a robust algorithm, based on a rho-coordinate expansion, to compute blocks and their derivatives efficiently around any diagonal point, with broad applicability to bootstrap analyses, including unequal external dimensions. The results provide both analytic insight into the analytic structure of blocks and practical tools for large-scale conformal bootstrap computations.

Abstract

Conformal blocks in any number of dimensions depend on two variables z, zbar. Here we study their restrictions to the special "diagonal" kinematics z = zbar, previously found useful as a starting point for the conformal bootstrap analysis. We show that conformal blocks on the diagonal satisfy ordinary differential equations, third-order for spin zero and fourth-order for the general case. These ODEs determine the blocks uniquely and lead to an efficient numerical evaluation algorithm. For equal external operator dimensions, we find closed-form solutions in terms of finite sums of 3F2 functions.