  We consider higher spin operators in weakly coupled gauge conformal field
theories. Crossing symmetry of mixed scalar correlators relates different
higher spin towers and we study the consequences for the spectrum and structure
constants of higher spin operators of different twists. Constraints are
obtained to all loops in perturbation theory. The large spin contributions to
the structure constants can be resummed into a theory-dependent prefactor times
a universal factor, whose structure of poles agrees with the one that would be
obtained from a Witten diagram supergravity computation, although only crossing
symmetry is assumed. Finally, our results provide an all loop expression for
the double null limit of mixed correlators, which is in perfect agreement with
the correlator/Wilson loop correspondence.
