Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT

TL;DR

This paper explores how reflection positivity can provide significant restrictions on conformal field theory data, complementing crossing symmetry constraints, and offers a new approach to bounding conformal dimensions and OPE coefficients.

Contribution

It demonstrates that reflection positivity alone can capture key restrictions usually derived from crossing symmetry in the bootstrap approach to CFTs.

Findings

Reflection positivity encodes constraints on OPE coefficients.

Certain functions related to CFT data are positive definite and completely monotonic.

The approach provides bounds on scalar conformal dimensions and OPE coefficients.

Abstract

Crossing symmetry (CS) is the main tool in the bootstrap program applied to CFT models. This consists in an equality which imposes restrictions on the CFT data of a model, i.e, the OPE coefficients and the conformal dimensions. Reflection positivity (RP) has also played a role, since this condition lead to the unitary bound and reality of the OPE coefficients. In this paper we show that RP can still reveal more information, showing how RP itself can capture an important part of the restrictions imposed by the full CS equality. In order to do that, we use a connection used by us in a previous work between RP and positive definiteness of a function of a single variable. This allows to write constraints on the OPE coefficients in a concise way, encoding in the conditions that certain functions of the crossratio will be positive defined and in particular completely monotonic. We will consider how the bounding of scalar conformal dimensions and OPE coefficients arise in this RP based approach. We will illustrate the conceptual and practical value of this view trough examples of general CFT models in $d$-dimensions.