Classical solutions by inverse scattering transformation in any number of dimensions. II. Instantons and large orders of the 1/N series for the (φ⃗²)² theory in ν dimensions (1≤ν≤4)

Instantons of the nonlocal effective action S_eff that generates a 1/N perturbative expansion for O(N)-symmetric (φ⃗²)² theory are obtained for Euclidean spatial dimension 0≤ν≤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 ν=0, 1, 2, 3, and 4, and 0≤g≤+∞. The large orders of the 1/N perturbative expansion are thus estimated. It is found that the 1/N perturbation series can be resummed by a Borel transform in integer dimension 0≤ν≤3. In four dimensions, the 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 (φ⃗²)² theory in four dimensions is nonperturbatively unstable. The saddle-point equation of massless (φ⃗²)² theory in the 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.