Summing Non-Borel Tunnelling Amplitudes by Variational Perturbation Theory

TL;DR

This paper introduces a method using variational perturbation theory to extract tunnelling amplitudes from divergent, non-Borel-summable perturbation series, accurately recovering instanton actions and higher loop effects.

Contribution

It develops an analytic continuation approach within variational perturbation theory to evaluate tunnelling amplitudes from divergent series, including instanton and fluctuation effects.

Findings

Accurately computes imaginary parts of partition functions and ground state energies.

Recovers instanton actions from divergent perturbation expansions.

Shows excellent agreement with exact known values.

Abstract

We present a method for extracting tunnelling amplitudes from perturbation expansions which are always divergent and not Borel-summable. We show that they can be evaluated by an analytic continuation of variational perturbation theory. The power of the method is illustrated by calculating the imaginary parts of the partition function of the anharmonic oscillator in zero spacetime dimensions and of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant $g$ and show that they are in excellent agreement with the exactly known values. As a highlight of the theory we recover from the divergent perturbation expansion of the tunnelling amplitude the action of the instanton and the effects of higher loop fluctuations around it.