  We derive an optical theorem for perturbative CFTs which computes the double
discontinuity of conformal correlators from the single discontinuities of lower
order correlators, in analogy with the optical theorem for flat space
scattering amplitudes. The theorem takes a purely multiplicative form in the
CFT impact parameter representation used to describe high-energy scattering in
the dual AdS theory. We use this result to study four-point correlation
functions that are dominated in the Regge limit by the exchange of the graviton
Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering is
dominated by double Pomeron exchange and receives contributions from tidal
excitations of the scattering states which are efficiently described by an AdS
vertex function, in close analogy with the known Regge limit result for
one-loop string scattering in flat space at finite string tension. We compare
the flat space limit of the conformal correlator to the flat space results and
thus derive constraints on the one-loop vertex function for type IIB strings in
AdS and also on general spinning tree level type IIB amplitudes in AdS.
