Anomalous dimensions in CFT with weakly broken higher spin symmetry

In a conformal field theory with weakly broken higher spin symmetry, the leading order anomalous dimensions of the broken currents can be efficiently determined from the structure of the classical non-conservation equations. We apply this method to the explicit example of $O(N)$ invariant scalar field theories in various dimensions, including the large $N$ critical $O(N)$ model in general $d$, the Wilson-Fisher fixed point in $d=4-\epsilon$, cubic scalar models in $d=6-\epsilon$ and the nonlinear sigma model in $d=2+\epsilon$. Using information from the $d=4-\epsilon$ and $d=2+\epsilon$ expansions, we obtain some estimates for the dimensions of the higher spin operators in the critical 3d $O(N)$ models for a few low values of $N$ and spin.