Asymptotically free Gauged-Yukawa systems

In this thesis we take a fresh look at asymptotic freedom for gauged-Yukawa models from a perspective that supersedes conventional studies within standard perturbation theory. In fact, our findings can be viewed as a generalization of previous investigations for non-Abelian Higgs models to systems which include a fermionic matter sector. We start with a minimalistic toy-model containing only a Yukawa and a QCD-like gauge sector. We then extend the analysis to a larger class of models which include the whole non-Abelian sector of the Standard Model. In both models we discover the existence of novel asymptotically free trajectories beyond standard perturbation theory. We construct such trajectories as quasi-fixed points for the Higgs scalar potential, whose couplings approach the noninteracting Gaußian fixed point with specific scalings with respect to the asymptotically free gauge couplings. We corroborate our findings in an effective-field-theory approach, and subsequently we obtain a comprehensive picture using the functional renormalization group. The latter method allows us to study the stability of the scalar potential for large field amplitudes. In contrast to standard perturbation theory, these new solutions become visible beyond the deep-Euclidean-regime, because of the important role of mass-threshold effects. Since one-loop universality is no longer guaranteed once threshold corrections are included, we investigate whether the existence of these ultraviolet complete trajectories is universal, i.e., a scheme-independent feature. We consider a wide class of regularization schemes that account for threshold behavior persisting in the infinite-energy limit, firstly focusing on the conventional minimal subtraction scheme and subsequently on mass-dependent schemes based on general momentum-space infrared regularizations. We argue that the existence of these asymptotically free solutions is a scheme-independent phenomenon.