Slightly broken higher-spin current in bosonic and fermionic QED in the large-$N$ limit

TL;DR

This paper investigates the anomalous dimensions of slightly broken higher-spin currents in various gauge-invariant conformal field theories, revealing a universal logarithmic asymptotic behavior at large spin that may be characteristic of gauge theories.

Contribution

It introduces a method to compute anomalous dimensions of higher-spin currents in gauge theories and uncovers a universal logarithmic asymptotic behavior at large spin.

Findings

Anomalous dimension scales as log s at large spin s.

Logarithmic behavior differs from non-gauge interacting CFTs.

Results apply to gauge theories and the $2+\epsilon$ dimensional WZW model.

Abstract

We study the slightly broken higher-spin currents in various CFTs with $\mathrm{U}(1)$ gauge field, including the tricritical QED, scalar QED, fermionic QED and QED-Gross-Neveu-Yukawa theory. We calculate their anomalous dimension by making use of the classical non-conservation equation and the equations of motion. We find a logarithmic asymptotic behaviour ($\gamma_s\sim 16/(N\pi^2) \log s $) of the anomalous dimension at large spin $s$, which is different from other interacting CFTs without gauge fields and may indicate certain unique features of gauge theories. We also study slightly broken higher-spin currents of the $\mathrm{SU}(N)_1$ WZW model at $d=2+\epsilon$ dimensions by formulating them as the QED theory, and we again find its anomalous dimension has a logarithmic asymptotic behaviour with respect to spin. This result resolves the mystery regarding the mechanism of breaking higher spin currents of Virasoro symmetry at $d=2+\epsilon$ dimensions, and may be applicable to other interesting problems such as the $2+\epsilon$ expansion of Ising CFT.