Conformal Bootstrap in Mellin Space

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 between certain observables in the 3D Ising model and the precise numerical values that have been recently obtained.