  We compute the four-loop beta functions of short and long-range multi scalar
models with general sextic interactions and complex fields. We then specialize
the beta functions to a $U(N)^3$ symmetry and study the renormalization group
at next-to-leading order in $N$ and small $\epsilon$. In the short-range case,
$\epsilon$ is the deviation from the critical dimension while it is the
deviation from the critical scaling of the free propagator in the long-range
case. This allows us to find the $1/N$ corrections to the rank-3 sextic tensor
model of arXiv:1912.06641. In the short-range case, we still find a non-trivial
real IR stable fixed point, with a diagonalizable stability matrix. All
couplings, except for the so-called wheel coupling, have terms of order
$\epsilon^0$ at leading and next-to-leading order, which makes this fixed point
different from the other melonic fixed points found in quartic models. In the
long-range case, the corrections to the fixed point are instead not
perturbative in $\epsilon$ and hence unreliable; we thus find no precursor of
the large-$N$ fixed point.
