# Closed-form expression for cross-channel conformal blocks near the lightcone

## TL;DR

This paper derives a closed-form expression for cross-channel conformal blocks near the lightcone in conformal field theories, using a new basis based on Lorentzian inversion, applicable to arbitrary spins and dimensions.

## Contribution

It introduces a natural basis for conformal blocks and provides a closed-form expression in terms of Kampé de Fériet functions for the lightcone limit.

## Key findings

- Derived a closed-form expression for conformal blocks near the lightcone.
- Identified a natural basis for cross-channel conformal blocks.
- Applied Lorentzian inversion to obtain results for identical external dimensions.

## Abstract

In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions of generic scalar primaries. Based on the nonperturbative pole structure in spin of Lorentzian inversion, we propose the natural basis functions for cross-channel conformal blocks. In this new basis, we find a closed-form expression for crossed conformal blocks in terms of the Kamp\'e de F\'eriet function, which applies to intermediate operators of arbitrary spin in general dimensions. We derive the general Lorentzian inversion for the case of identical external scaling dimensions. Our results for the lightcone limit also shed light on the complete analytic structure of conformal blocks in the lightcone expansion.