Two and four-loop $\beta$-functions of rank 4 renormalizable tensor field theories

A recent rank 4 tensor field model generating 4D simplicial manifolds has been proved to be renormalizable at all orders of perturbation theory [arXiv:1111.4997 [hep-th]]. The model is built out of $\phi^6$ ($\phi^6_{(1/2)}$), $\phi^4$ ($\phi^4_{(1)}$) interactions and an anomalous term ($\phi^4_{(2)}$). The $\beta$-functions of this model are evaluated at two and four loops. We find that the model is asymptotically free in the UV for both the main $\phi^6_{(1/2)}$ interactions whereas it is safe in the $\phi^4_{(1)}$ sector. The remaining anomalous term turns out to possess a Landau ghost.