Perturbation theory at large order. II. Role of the vacuum instability

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Abstract
We extend our previous study of large orders of perturbation series for nonrelativistic quantum mechanics and boson field theories to more complicated situations. It is shown that when perturbation theory is performed around an unstable vacuum and does not reveal any pathology at low orders the existence of real pseudoparticles, which are responsible for the tunneling to a more stable vacuum, also implies the divergence and the non-Borel-summability of the series. Conversely, large orders of perturbations around a stable vacuum are dominated by complex solutions to Euclidean field equations. They quantitatively characterize its behavior and indicate the Borel summability of the series. Thus the corresponding Green's functions are unambiguously determined by their perturbation series.