Bulk Landau Pole and Unitarity of Dual Conformal Field Theory

The singlet sector of the $O(N)$ $\phi^4$-model in AdS$_4$ at large-$N$, gives rise to a dual conformal field theory on the conformal boundary of AdS$_4$, which is a deformation of the generalized free field. We identify and compute an AdS$_4$ three-point one-loop fish diagram that controls the exact large-$N$ dimensions and operator product coefficients (OPE) for all ``double trace'' operators as a function of the renormalized $\phi^4$-couplings. We find that the space of $\phi^4$-coupling is compact with a boundary at the bulk Landau pole. The dual CFT is unitary only in an interval of ${\it negative}$ couplings bounded by the Landau pole where the lowest OPE coefficient diverges.