  Double trace deformations, that is products of two local operators, define
perturbations of conformal field theories that can be studied exactly in the
large-N limit. Even when the double trace deformation is irrelevant in the
infrared, it is believed to flow to an ultraviolet fixed point. In this note we
define the Kallen-Lehmann representation of the two-point function of a local
operator O in a theory perturbed by the square of such operator. We use such
representation to discover potential pathologies at intermediate points in the
flow that may prevent to reach the UV fixed point. We apply the method to an
"extremal" deformation that naively would flow to a UV fixed point where the
operator O would saturate the unitarity bound. We find that the UV fixed point
is not conformal and that the deformed two-point function propagates unphysical
modes. We interpret the result as showing that the flow to the UV fixed point
does not exist. This resolves a potential puzzle in the holographic
interpretation of the deformation.
