The old conformal bootstrap revisited

TL;DR

This paper revisits the old conformal bootstrap method, applying advanced multi-loop Feynman integral techniques to solve for conformal dimensions in specific theories, including $^3$ in 7D and $O(N)$ models in 3D.

Contribution

It introduces an efficient approach to solving the old conformal bootstrap equations using multi-loop Feynman integrals, providing solutions at first order in the skeleton expansion.

Findings

Solutions for conformal dimensions at finite coupling and integer dimensions.

Application to $^3$ theory in seven dimensions.

Application to $O(N)$ vector models in three dimensions.

Abstract

The "old" conformal bootstrap was originally formulated by Migdal and Polyakov (MP) as a method for calculating conformal dimensions self-consistently. In this work we revisit the MP bootstrap and apply efficient multi-loop Feynman integral techniques in order to solve the corresponding equations. We obtain solutions at first order in the skeleton expansion for finite coupling and for integer values of the spacetime dimension. We focus in particular on the $\phi^3$ theory in seven dimensions and the $O(N)$ vector models in three dimensions.