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)

Abstract

Instantons of the nonlocal effective action ${S}_{\mathrm{eff}}$ that generates a $\frac{1}{N}$ perturbative expansion for $\mathrm{O}(N)$-symmetric ${({\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}}^{2})}^{2}$ theory are obtained for Euclidean spatial dimension $0\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}4$, 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 $\ensuremath{\nu}=0, 1, 2, 3, \mathrm{and} 4$, and $0\ensuremath{\le}g\ensuremath{\le}+\ensuremath{\infty}$. 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 $0\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}3$. 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 ${({\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}}^{2})}^{2}$ theory in four dimensions is nonperturbatively unstable. The saddle-point equation of massless ${({\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}}^{2})}^{2}$ 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.