Radial expansion for spinning conformal blocks

TL;DR

The paper presents a unified radial (Hogervorst–Rychkov) expansion framework to construct bosonic conformal blocks with external spin in any dimension. It derives closed-form recurrence relations for expansion coefficients by reformulating Casimir equations in a Gegenbauer basis and, separately, by leveraging the analytic structure in the exchanged dimension elta (Zamolodchikov-like recursion). The authors demonstrate the method on several spinning cases (one vector, two vectors, and two spin-2 external operators) and provide explicit recursion data, initial conditions, and illustrative level-by-level coefficients. This approach yields a practical, efficient route for numerically implementing spinning conformal blocks in bootstrap computations and clarifies the connections between block structure and representation theory, including seed-block strategies and potential extensions to conserved operators.

Abstract

This paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the known result when the external operators are all the same scalar operator, but it allows to compute conformal blocks for external operators with spin. Moreover, we explain how to write closed form recursion relations for the coefficients of the expansions. We study three examples of four point functions in detail: one vector and three scalars; two vectors and two scalars; two spin 2 tensors and two scalars. Finally, for the case of two external vectors, we also provide a more efficient way to generate the series expansion using the analytic structure of the blocks as a function of the scaling dimension of the exchanged operator.

Figures (7)

• Figure 1: Cylinder configuration that leads to the radial coordinates of Hogervorst:2013sma.
• Figure 2: Support of the coefficients $w(m,j)$ in the expansion of ${\mathcal{G}}_{\Delta, l}$.
• Figure 3: The set of points ${\mathcal{S}}$ involved in the recurrence relation for the coefficients $w(m,j)$ of the scalar conformal block.
• Figure 4: The sets ${\mathcal{S}}^s_{s\rq{}}$ appearing in the linear combination (\ref{['recrel1Vector']}) for $s=1,2$. Red circles correspond to $s'=2$ and black dots correspond to $s'=1$.
• Figure 5: Pictorial representation of the sets ${\mathcal{S}}^s_{s\rq{}}$ in formula (\ref{['recrel2Vectors']}). Increasing values of $s\rq{}$ correspond to increasing radius of the circles.