Sphalerons and Large Order Behavior of Perturbation Theory in Lower Dimension

Sphalerons – unstable static solutions of classical field equations in (d+1)-dimensional space-time – may be viewed as euclidean solutions in d dimensions. We discuss their role in the large order asymptotics of the perturbation theory. Specifically, we calculate their contribution to the large order behaviour of the ground state energy in a quantum mechanical model. When the number of negative modes is odd, single sphaleron contribution dominates, while this contribution vanishes when the number of negative modes is even. These results are confirmed by numerical calculations.