2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/211222The $β$-functions of O(N) and U(N) invariant Grosse-Wulkenhaar models are computed at one loop using the matrix basis. In particular, for ``parallel interactions", the model is proved asymptotically free in the UV limit for $N >1$, and has a triviality problem or Landau ghost for $N<1$. The vanishing $β$-function is recovered solely at N=1. We discuss various possible consequences of these results.14 pages, 6 FiguresHigh Energy Physics - Theorytext