Scalar-Vector Bootstrap

TL;DR

This paper develops the detailed framework for applying the conformal bootstrap to four-point functions involving two scalars and two vectors, including tensor structures, conformal blocks, and crossing symmetry equations.

Contribution

It provides a comprehensive construction of conformal blocks and bootstrap equations for mixed scalar-vector correlators in arbitrary dimensions, with special focus on conserved vectors.

Findings

Explicit conformal blocks as differential operators on scalar blocks

Complete set of bootstrap equations for scalar-vector four-point functions

Simplifications for conserved vector cases

Abstract

We work out all of the details required for implementation of the conformal bootstrap program applied to the four-point function of two scalars and two vectors in an abstract conformal field theory in arbitrary dimension. This includes a review of which tensor structures make appearances, a construction of the projectors onto the required mixed symmetry representations, and a computation of the conformal blocks for all possible operators which can be exchanged. These blocks are presented as differential operators acting upon the previously known scalar conformal blocks. Finally, we set up the bootstrap equations which implement crossing symmetry. Special attention is given to the case of conserved vectors, where several simplifications occur.