  Asymptotically free renormalization group trajectories can be constructed in
nonabelian Higgs models with the aid of generalized boundary conditions imposed
on the renormalized action. We detail this construction within the languages of
simple low-order perturbation theory, effective field theory, as well as modern
functional renormalization group equations. We construct a family of explicit
scaling solutions using a controlled weak-coupling expansion in the
ultraviolet, and obtain a standard Wilsonian RG relevance classification of
perturbations about scaling solutions. We obtain global information about the
quasi-fixed function for the scalar potential by means of analytic asymptotic
expansions and numerical shooting methods. Further analytical evidence for such
asymptotically free theories is provided in the large-N limit. We estimate the
long-range properties of these theories, and identify initial/boundary
conditions giving rise to a conventional Higgs phase.
