A Manual for Conformal Field Theories in 4D

This thesis conducts an investigation of four dimensional conformal field theories (CFT). With the application of the bootstrap techniques in mind, we set out to compute the conformal partial waves (CPW) needed to bootstrap a generic four-point function in a 4D CFT. These CPW can correspond to the exchange of a bosonic or fermionic operator, in an irreducible representation $(\ell,\bar\ell)$ of the Lorentz group. Utilizing the embedding formalism in twistor space, we introduce a basis of differential operators that can relate any CPW to a ``seed'' CPW with the same exchanged operator. We compute in a closed analytic form the seed CPW. We solve the Casimir equations by using an educated ansatz and reduce the problem to an algebraic linear system. Many of the properties of the ansatz are deduced using the shadow formalism.The seed CPW depends on the representation of the exchanged operator, particularly on the value of $p=|\ell-\bar\ell|$ and its complexity grows with $p$. As an application of our results, we write the bootstrap equations for a four-point function of two scalar and two spin-1/2 fermions. We solve the equations in the light-cone limit and compute the anomalous dimensions of double-twist operators as an expansion in 1/spin in the large spin limit.