  We explore O(N) models in dimensions $4<d<6$. Specifically, we investigate
models of an O(N) vector field coupled to an additional scalar field via a
cubic interaction. Recent results in $d=6-\epsilon$ have uncovered an
interacting ultraviolet fixed point of the renormalization group (RG) if the
number N of components of the vector field is large enough, suggesting that
these models are asymptotically safe. We set up a functional RG analysis of
these systems to address three key issues: Firstly, we find that in $d=5$ the
interacting fixed point exists all the way down to N=1. Secondly, we show that
the standard O(N) universality classes are actually embedded in those of the
cubic models, in that the latter exhibit the same values for (most of) the
critical exponents, but feature an additional third RG relevant direction.
Thirdly, we address the critical question of global stability of the
fixed-point potential to test whether the fixed point can underly a viable
quantum field theory.
