On critical exponents without Feynman diagrams

TL;DR

This paper revisits Polyakov's 1974 conformal bootstrap approach, avoiding Feynman diagrams, to analytically compute anomalous dimensions of operators in the $O(n)$ model at the Wilson-Fisher fixed point up to second order in epsilon.

Contribution

It extends Polyakov's method to calculate anomalous dimensions in the $O(n)$ model without Feynman diagrams, providing a new analytical tool for conformal bootstrap studies.

Findings

Computed anomalous dimensions up to $O(psilon^2)$ in the $O(n)$ model.

Demonstrated the effectiveness of a diagram-free bootstrap approach.

Extended the analytical understanding of the Wilson-Fisher fixed point.

Abstract

In order to achieve a better analytic handle on the modern conformal bootstrap program, we re-examine and extend the pioneering 1974 work of Polyakov's, which was based on consistency between the operator product expansion and unitarity. As in the bootstrap approach, this method does not depend on evaluating Feynman diagrams. We show how this approach can be used to compute the anomalous dimensions of certain operators in the $O(n)$ model at the Wilson-Fisher fixed point in $4-\epsilon$ dimensions up to $O(\epsilon^2)$.