A Note on Instantons in a 1D Same-Level Asymmetric Double Well

TL;DR

This paper derives formulas for multi-instanton corrections in a 1D asymmetric double well, accounting for different curvatures at the wells, and extends Coleman's formulas to this more general case.

Contribution

It provides new formulas for instanton corrections in asymmetric double wells with different curvatures, expanding on Coleman's original work.

Findings

Formulas for multi-instanton corrections derived

Extension of Coleman's formulas to asymmetric potentials

Applications to symmetric and triple well examples

Abstract

We prove formulas for the multi-instanton corrections to the overlap and energies of a 1D same-level asymmetric double well using the Euclidean path integral. Both the odd and even instanton sectors are summed to all orders. The double well is same-level asymmetric in the sense that the potentials at neighboring wells have the same bottom level but can have different Hessians/curvatures/frequencies, which modify Coleman's original formulas. This for instance implies that the reference model used to calculate the functional determinant of quantum fluctuations must now interpolate between simple harmonic oscillators of different frequencies. Examples of symmetric double and triple wells are worked out.