Superconformal Symmetry, Correlation Functions and the Operator Product Expansion

Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When $\N=4$, results are obtained for the three point function of various descendant operators, including the energy momentum tensor and SU(4) current. For both $\N=2$ or 4 superconformal identities are derived for the functions of the two conformal invariants appearing in the four point function for the chiral primary operator. These are solved in terms of a single arbitrary function of the two conformal invariants and one or three single variable functions. The results are applied to the operator product expansion using the exact formula for the contribution of an operator in the operator product expansion in four dimensions to a scalar four point function. Explicit expressions representing exactly the contribution of both long and possible short supermultiplets to the chiral primary four point function are obtained. These are applied to give the leading perturbative and large N corrections to the scale dimensions of long supermultiplets.