Color Grosse–Wulkenhaar models: one-loop β-functions

The 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 to be asymptotically free in the UV limit for N>1, and it 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.