Fixed points of the renormalisation group running of quark and fermion mixing matrices in the Standard Model and beyond

The renormalisation group running of fermion mixing matrices in the Standard model and beyond is studied. For the massless 1-loop running with three generations six fixed points are found. Their associated anomalous dimension matrices are calculated and the nature of each fixed point, whether attractive, repulsive or mixed, is determined. An argument is given that the fixed points found at 1-loop must remain fixed points to all orders in perturbation theory and even non-perturbatively, as they are associated with certain differential geometric properties of vector fields on the space of mixing matrices. With $N_g$ dark or sterile neutrinos there are at least $N_g!$ fixed points of the fermion mixing matrix.