The taste of supersymmetry in critical Yukawa models: from one to four flavours

In this thesis we renormalized a general Yukawa theory that depends on an arbitrary number of fields known as flavours up to 1-loop using the Background field method and without imposing any symmetry. Using numerical methods, we found the $\beta$-functions' fixed points. Among these, we found SUSY invariant fixed points for various number of flavours, $N_f$. These supersymmetric fixed points are protected by a special parity transformations because relevant SUSY breaking operators are not parity invariant. With the results at hand, we classified the theories by their symmetries and obtained the anomalous dimensions and their stability matrices. Several interesting results have been found: first of all, we found the complete set of solutions of the $\beta$-function for $N_f=2,3$. Furthermore, a new, yet potentially partial (due to computational limitations) set of supersymmetric fixed points was found for $N_f=4$. Since the Yukawa theory is a bounded (from below) theory, we expected to discover only perturbatively unitary theories and, indeed, this is what we found