Solving CFTs with Weakly Broken Higher Spin Symmetry

TL;DR

This paper introduces a novel approach using large spin perturbation theory to analyze conformal field theories with weakly broken higher spin symmetry, deriving spectra from crossing symmetry without relying on Lagrangian formulations.

Contribution

The authors develop a method to determine the spectrum of broken currents in weakly broken higher spin symmetric CFTs directly from crossing symmetry, applicable to various models including gauge theories.

Findings

Spectrum of broken currents derived for any spin

Reproduces known spectra of several models like Wilson-Fisher and large N models

Provides an on-shell gauge invariant approach to gauge theories

Abstract

The method of large spin perturbation theory allows to analyse conformal field theories (CFT) by turning the crossing equations into an algebraic problem. We apply this method to a generic CFT with weakly broken higher spin (HS) symmetry, to the first non-trivial order in the breaking parameter. We show that the spectrum of broken currents, for any value of the spin, follows from crossing symmetry. After discussing a generic model of a single scalar field, we focus on vector models with $O(N)$ global symmetry. We rediscover the spectrum of several models, including the $O(N)$ Wilson-Fisher model around four dimensions, the large $O(N)$ model in $2<d<4$ and cubic models around six dimensions, not necessarily unitary. We also discuss models where the fundamental field is not part of the spectrum. Examples of this are weakly coupled gauge theories and our method gives an on-shell gauge invariant way to study them. At first order in the coupling constant we show that again the spectrum follows from crossing symmetry, to all values of the spin. Our method provides an alternative to usual perturbation theory without any reference to a Lagrangian.