Classical solutions by inverse scattering transformation in any number of dimensions. II. Instantons and large orders of the1Nseries for the(φ→2)2theory inνdimensions (1≤ν≤4)

Instantons of the nonlocal effective action $S_{\mathrm{eff}}$ that generates a $\frac{1}{N}$ perturbative expansion for $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric ${\left({\overrightarrow{\phi }}^2\right)}^2$ theory are obtained for Euclidean spatial dimension $0\le \nu \le 4$, through the inverse scattering transformation (IST). They are studied analytically to a large extent. In addition, variational methods are used when the IST does not provide a closed solution for all couplings. The values of the instanton action are given as a function of the coupling constant $g$ for $\nu =0, 1, 2, 3, \mathrm{and} 4$, and $0\le g\le +\infty$. The large orders of the $\frac{1}{N}$ perturbative expansion are thus estimated. It is found that the $\frac{1}{N}$ perturbation series can be resummed by a Borel transform in integer dimension $0\le \nu \le 3$. In four dimensions, the $\frac{1}{N}$ perturbation series is not Borel-summable, owing to the existence of an instanton with real positive action, for physically relevant values of the renormalized coupling constant. It is concluded that ${\left({\overrightarrow{\phi }}^2\right)}^2$ theory in four dimensions is nonperturbatively unstable. The saddle-point equation of massless ${\left({\overrightarrow{\phi }}^2\right)}^2$ theory in the $\frac{1}{N}$ expansion is found to be completely integrable at least for spherically symmetric fields. Explicit instanton solutions are given for this case. A large-$N$ estimate of the decay rate of the vacuum is given.