  We develop the idea of an effective conformal theory describing the low-lying
spectrum of the dilatation operator in a CFT. Such an effective theory is
useful when the spectrum contains a hierarchy in the dimension of operators,
and a small parameter whose role is similar to that of 1/N in a large N gauge
theory. These criteria insure that there is a regime where the dilatation
operator is modified perturbatively. Global AdS is the natural framework for
perturbations of the dilatation operator respecting conformal invariance, much
as Minkowski space naturally describes Lorentz invariant perturbations of the
Hamiltonian. Assuming that the lowest-dimension single-trace operator is a
scalar, O, we consider the anomalous dimensions, gamma(n,l), of the
double-trace operators of the form O (del^2)^n (del)^l O. Purely from the CFT
we find that perturbative unitarity places a bound on these dimensions of
|gamma(n,l)|<4. Non-renormalizable AdS interactions lead to violations of the
bound at large values of n. We also consider the case that these interactions
are generated by integrating out a heavy scalar field in AdS. We show that the
presence of the heavy field "unitarizes" the growth in the anomalous
dimensions, and leads to a resonance-like behavior in gamma(n,l) when n is
close to the dimension of the CFT operator dual to the heavy field. Finally, we
demonstrate that bulk flat-space S-matrix elements can be extracted from the
large n behavior of the anomalous dimensions. This leads to a direct connection
between the spectrum of anomalous dimensions in d-dimensional CFTs and
flat-space S-matrix elements in d+1 dimensions. We comment on the emergence of
flat-space locality from the CFT perspective.
