Case $d\geq 5$:

The result is not only perfectly regular, but the second term in the right-hand side vanishes in the limit, so long as $\lambda$ is kept finite, and one is left with the simple result $J_{0}=m_{\mathfrak{N}}^{2}V_{0}$, which is consistent with a trivial theory. In this case this result is valid for any finite value of $\lambda$. This is consistent with the triviality of the model for $d\geq 5$, which seems to be a fairly well-established fact.

It is interesting to observe that it is possible to define a version of this model in which the limit $\lambda\to\infty$ is taken. It is possible to show, with all mathematical rigor, and for any $d$ and any $\mathfrak{N}$, that the limit of the $SO(\mathfrak{N})$ polynomial model we have here, in which one makes $\lambda\to\infty$ and $\alpha\to-\infty$ in such a way that $\beta=-\alpha/\lambda$ is kept finite, is in fact the $SO(\mathfrak{N})$ non-linear Sigma Model with coupling constant $\beta$. It is therefore possible that the $SO(\mathfrak{N})$ non-linear Sigma Models in $d=5$ or more may still have some interesting continuum limits.