Scalar-Vector Bootstrap

TL;DR

The paper develops the conformal bootstrap for a four-point function of two scalars and two vectors in a general CFT in $D$ dimensions. It constructs full spinning conformal blocks by representing exchange blocks (scalar, vector, and mixed-symmetry) as differential operators acting on scalar blocks, using the shadow formalism in embedding space. It provides explicit tensor structures, embedding-space projectors, and mixing/shadow matrices, and sets up crossing-symmetric bootstrap equations with special simplifications for conserved vectors. This framework enables analytic and numerical bounds on the operator spectrum and OPE data in theories with global symmetries and has potential connections to holography.

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.