Anomalous dimensions in CFT with weakly broken higher spin symmetry
Simone Giombi, Vladimir Kirilin
TL;DR
The paper develops a general, loop-free method to compute leading-order anomalous dimensions γ_s of weakly broken higher-spin currents in CFTs with weak higher-spin symmetry, by relating γ_s to the classical non-conservation equations ∂·J_s = g K_{s-1} and evaluating tree-level ⟨K_{s-1} K_{s-1}⟩ relative to ⟨J_s J_s⟩. It applies the method to O(N) vector models across dimensions: the large-N critical O(N) model, the Wilson-Fisher fixed point in d=4−ε, cubic models in d=6−ε, and the nonlinear sigma model in d=2+ε, obtaining explicit expressions for γ_s in singlet, symmetric, and antisymmetric sectors, reproducing known results and deriving new mixing structures in the cubic case and singlet sectors in the sigma model. The work also analyzes large-spin and asymptotic behavior, confirms AdS/CFT expectations for higher-spin duals, and provides two-sided Padé estimates to extrapolate γ_s to d=3, along with discussion of non-unitary fixed points and finite-N corrections. The approach offers a unified framework to connect conformal symmetry, higher-spin constraints, and perturbative expansions across dimensions.
Abstract
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 $d=4-ε$, cubic scalar models in $d=6-ε$ and the nonlinear sigma model in $d=2+ε$. Using information from the $d=4-ε$ and $d=2+ε$ 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.