Exact Correlation Functions in $SU(2)$ $\mathcal{N}=2$ Superconformal QCD

We report an exact solution of 2- and 3-point functions of chiral primary fields in $SU(2)$ $\mathcal{N}=2$ super-Yang-Mills theory coupled to four hypermultiplets. It is shown that these correlation functions are nontrivial functions of the gauge coupling, obeying differential equations which take the form of the semi-infinite Toda chain. We solve these equations recursively in terms of the Zamolodchikov metric that can be determined exactly from supersymmetric localization on the four-sphere. Our results are verified independently in perturbation theory with a Feynman diagram computation up to 2 loops. This is a short version of a companion paper that contains detailed technical remarks, additional material, and aspects of an extension to the $SU(N)$ gauge group.