Generalized Wilson-Fisher critical points from the conformal OPE

TL;DR

The paper develops a method to analyze smooth deformations of Generalized Free CFTs by exploiting the singularity structure of conformal blocks arising from null states, enabling the computation of leading anomalous dimensions in an $\\epsilon$-expansion without employing equations of motion. It applies this framework to generalized multicritical ($\\phi^{2n}$) points and $O(N)$ models across dimensions, deriving explicit expressions for leading ($\\gamma^{(1)}$) and next-to-leading ($\\gamma^{(2)}$) anomalous dimensions and recovering known Wilson-Fisher results in canonical cases. The approach offers a unified perturbative scheme that does not depend on crossing symmetry or unitarity and can potentially address non-unitary critical systems, with results aligning with established methods where overlap exists. The work points to broader applicability, including odd deformations and spinning operators, and is complemented by a longer detailed version anticipated elsewhere.

Abstract

We study possible smooth deformations of Generalized Free Conformal Field Theories in arbitrary dimensions by exploiting the singularity structure of the conformal blocks dictated by the null states. We derive in this way, at the first non trivial order in the $ε$-expansion, the anomalous dimensions of an infinite class of scalar local operators, without using the equations of motion. In the cases where other computational methods apply, the results agree.