  We continue the study of model-independent constraints on the unitary
Conformal Field Theories in 4-Dimensions, initiated in arXiv:0807.0004. Our
main result is an improved upper bound on the dimension \Delta of the leading
scalar operator appearing in the OPE of two identical scalars of dimension d.
In the interval 1<d<1.7 this universal bound takes the form
\Delta<2+0.7(d-1)^{1/2}+2.1(d-1)+0.43(d-1)^{3/2}. The proof is based on prime
principles of CFT: unitarity, crossing symmetry, OPE, and conformal block
decomposition. We also discuss possible applications to particle phenomenology
and, via a 2-D analogue, to string theory.
