Fixed Points in Supersymmetric Extensions of the Standard Model

We search for weakly interacting fixed points in extensions of the minimally supersymmetric standard model (MSSM). Necessary conditions lead to three distinct classes of anomaly-free extensions involving either new quark singlets, new quark doublets, or a fourth generation. While interacting fixed points arise prolifically in asymptotically free theories, their existence is significantly constrained as soon as some of the non-abelian gauge sectors are infrared free. Performing a scan over $\sim 200$k different MSSM extensions using matter field multiplicities and the number of superpotential couplings as free parameters, we find mostly infrared conformal fixed points, and a small subset with ultraviolet ones. All settings predict low-scale supersymmetry-breaking and a violation of $R$-parity. Despite of residual interactions, the running of couplings out of asymptotically safe fixed points is logarithmic as in asymptotic freedom. Some fixed points can be matched to the Standard Model though the matching scale comes out too low. Prospects for higher matching scales and asymptotic safety beyond the MSSM are indicated.