Large-Nbehavior for independent-value models

Highly specialized, $N$-component, scalar model quantum field theories invariant under $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ transformations are studied in the limit of large $N$. The models are expressd in $n$-dimensional Euclidean space-time and differ from conventional covariant quantum models by the absence of all space-time gradients, a modification that leads to nonrenormalizable $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-invariant interactions for each $N\ge 1$. These models are solved by nonperturbative techniques, and the solutions exhibit two striking and unfamiliar properties: (1) For finite (or infinite) $N$, the solutions of any interacting theory do not reduce to those of the free theory in the limit where the coupling of the nonlinear interaction vanishes; and (2) the relevant (asymptotic) dependence of the parameters of the interacting theories on $N$ differs from the conventional choice, and the limit $N\to \infty$ does not lead to a Hartree-type solution. It is proposed that similar unconventional behavior may characterize certain $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-invariant, covariant nonrenormalizable quantum field theories, and in particular that the limit $N\to \infty$ may not lead to a Hartree (or Hartree-Fock) type of solution.