Bootstrapping Veneziano Amplitude of Vasiliev Theory and $3D$ Bosonization

Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the four-point functions at arbitrary 't Hooft coupling $\lambda$ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the $\lambda$ expansions of the correlation functions. The $\lambda$ expansions terminate at order $\lambda^2$ and the solutions are exact in $\lambda$. Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the $3D$ Chern-Simons vector models and the Vasiliev theory in $AdS_4$. Besides, we show that the same results can also be obtained rather simply from bosonization duality of $3D$ Chern-Simons vector models. The odd term at order $O(\lambda)$ in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the $3D$ bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.