  We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study
the effects caused by its metastability. As shown in previous work, this model
formally possesses a fixed point where, perturbatively in the $1/N$ expansion,
the operator scaling dimensions are real and above the unitarity bound. Here,
we further show that these scaling dimensions do acquire small imaginary parts
due to the instanton effects. In $d$ dimensions and for large $N$, we find that
they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals
the sphere free energy of a conformal scalar in $d-2$ dimensions. The
non-perturbatively small imaginary parts also appear in other observables, such
as the sphere free energy and two and three-point function coefficients, and we
present some of their calculations. Therefore, at sufficiently large $N$, the
$O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large
enough for the imaginary parts to be numerically negligible, the
five-dimensional $O(N)$ models may be studied using the techniques of numerical
bootstrap.
