Deconstructing Conformal Blocks in 4D CFT

TL;DR

The paper develops a differential-operator framework to relate conformal partial waves for tensor correlators in 4D CFTs, enabling any CPW to be generated from a small set of seed correlators. It leverages the 6D embedding formalism in twistor space to classify three-point functions and to build a differential basis that maps between structures, then extends to mixed-symmetry exchanges and general four-point tensor structures. By introducing seed CPWs, particularly for traceless-symmetric external legs, the authors show how the full tensor CPW problem can be reduced to a finite set of scalar-like seeds, dramatically simplifying tensor bootstrap computations. The work includes detailed recursion relations, parity considerations, and concrete examples (four-fermion and conserved-current cases), and outlines the remaining steps to compute seed CPWs and achieve a complete 4D tensor bootstrap framework with wide practical impact.

Abstract

We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few "seed" correlators, simplifying dramatically the computation needed to bootstrap tensor correlators.