  Enhanced tensor field theories (eTFT) have dominant graphs that differ from
the melonic diagrams of conventional tensor field theories. They therefore
describe pertinent candidates to escape the so-called branched polymer phase,
the universal geometry found for tensor models. For generic order $d$ of the
tensor field, we compute the perturbative $\beta$-functions at one-loop of two
just-renormalizable quartic eTFT coined by $+$ or $\times$, depending on their
vertex weights. The models $+$ has two quartic coupling constants $(\lambda,
\lambda_{+})$, and two 2-point couplings(mass, $Z_a$). Meanwhile, the model
$\times$ has two quartic coupling constants $(\lambda, \lambda_{\times})$ and
three 2-point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models
have a constant wave function renormalization: $Z=1$ and therefore no anomalous
dimension. Despite such peculiar behavior, both models acquire nontrivial
radiative corrections for the coupling constants. The RG flow of the model $+$
exhibits a particular asymptotic safety: $\lambda_{+}$ is marginal without
corrections thus is a fixed point of arbitrary constant value. All remaining
couplings determine relevant directions and get suppressed in the UV.
Concerning the model $\times$, $\lambda_{\times}$ is marginal and again a fixed
point (arbitrary constant value), $\lambda$, $\mu$ and $Z_a$ are all relevant
couplings and flow to 0. Meanwhile $Z_{2a}$ is a marginal coupling and becomes
a linear function of the time scale. This model can neither be called
asymptotically safe or free.
