# Recursion Relations in Witten Diagrams and Conformal Partial Waves

## TL;DR

This paper develops a recursive method for decomposing Witten diagrams into conformal blocks in various channels, revealing infinite relations among coefficients and simplifying calculations in conformal field theories.

## Contribution

It introduces a recursive algorithm for conformal block decomposition of exchange Witten diagrams, utilizing linear relations among coefficients, applicable in both one and higher dimensions.

## Key findings

- Derived infinite linear relations among decomposition coefficients.
- Formulated a recursive algorithm for coefficients in 1D and higher-dimensional CFTs.
- Provided a recursive approach for conformal partial wave decomposition.

## Abstract

We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.