Masses of higher spin fields on ${\mathrm{AdS}}_4$ and conformal perturbation theory

We study the breaking of gauge symmetry for higher spin theory on ${\mathrm{AdS}}_4$ dual to the 3d critical $\mathrm{O}(N)$ vector model. It was argued that the breaking is due to the change of boundary condition for a scalar field through a loop effect and the Goldstone modes are bound states of a scalar field and higher spin field. The masses of higher spin fields were obtained from the anomalous dimensions of dual currents at the leading order in $1/N$, and we reproduce them from the $\mathrm{O}(N)$ vector model in the conformal perturbation theory. The anomalous dimensions can be computed from the bulk theory using Witten diagrams, and we show that the bulk computation reduces to the boundary one in the conformal perturbation theory. With this fact our computation provides an additional support for the bulk interpretation.

Figure 1

The Witten diagram corresponding to the one-loop contribution of current-current two point function.

Figure 2

The Witten diagrams with $(m+n-2)$ insertions of boundary deformation operator.

Figure 3

The Witten diagrams corresponding to the $f^2$ order contributions to the current-current two point function.

Figure 4

The effective Witten diagrams corresponding to the integrals we need to compute. The dual scaling dimensions of bulk scalar propagating along the lines are added.