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Two and four-loop $β$-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 $ϕ^6$ ($ϕ^6_{(1/2)}$), $ϕ^4$ ($ϕ^4_{(1)}$) interactions and an anomalous term ($ϕ^4_{(2)}$). The $β$-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 $ϕ^6_{(1/2)}$ interactions whereas it is safe in the $ϕ^4_{(1)}$ sector. The remaining anomalous term turns out to possess a Landau ghost.