# Recursion relation for general 3d blocks

## TL;DR

This paper derives explicit formulas for recursion relations of 3D spinning conformal blocks, facilitating efficient computation crucial for numerical conformal bootstrap applications.

## Contribution

It provides the first closed-form expressions for all ingredients of the recursion relation for general spinning conformal blocks in 3D, enabling automatic generation of conformal block tables.

## Key findings

- Closed-form expressions for recursion ingredients in 3D
- Path to efficient conformal block table generation
- Generalization to higher dimensions d

## Abstract

We derive closed-form expressions for all ingredients of the Zamolodchikov-like recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a path to efficient automatic generation of conformal block tables, which has immediate applications in numerical conformal bootstrap program. Our derivation is based on an understanding of null states and conformally-invariant differential operators in momentum space, combined with a careful choice of the relevant tensor structures bases. This derivation generalizes straightforwardly to higher spacetime dimensions d, provided the relevant Clebsch-Gordan coefficients of Spin(d) are known.