  We propose a new approach towards analytically solving for the dynamical
content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This
combines the original bootstrap idea of Polyakov with the modern technology of
the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams
with built in crossing symmetry as our basic building blocks rather than the
conventional conformal blocks in a particular channel. Demanding consistency
with the operator product expansion (OPE) implies an infinite set of
constraints on operator dimensions and OPE coefficients. We illustrate the
power of this method in the epsilon expansion of the Wilson-Fisher fixed point
by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients
to higher orders in epsilon than currently available using other analytic
techniques (including Feynman diagram calculations). Our results enable us to
get a somewhat better agreement of certain observables in the 3d Ising model,
with the precise numerical values that have been recently obtained.
