The euclidean propagator in a model with two non-equivalent instantons

TL;DR

This paper analyzes quantum tunneling in a complex octic potential using the Euclidean propagator and semiclassical instanton methods, accounting for fluctuations and multi-instanton effects to understand tunneling phenomena.

Contribution

It provides a detailed semiclassical analysis of tunneling in a model with two non-equivalent instantons, including the computation of functional determinants and multi-instanton contributions.

Findings

Explicit expression for the Euclidean propagator in the model

Analysis of the stability equation and zero-mode behavior

Inclusion of multi-instanton effects via dilute-gas approximation

Abstract

We consider in detail how the quantum-mechanical tunneling phenomenon occurs in a well-behaved octic potential. Our main tool will be the euclidean propagator just evaluated between two minima of the potential at issue. For such a purpose we resort to the standard semiclassical approximation which takes into account the fluctuations over the instantons, i.e. the finite-action solutions of the euclidean equation of motion. As regards the one-instanton approach, the functional determinant associated with the so-called stability equation is analyzed in terms of the asymptotic behaviour of the zero-mode. The conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the two different frequencies derived from the minima of the potential involved in the computation. The second instanton of the model is studied in a similar way. The physical effects of the multi-instanton configurations are included in this context by means of the alternate dilute-gas approximation where the two instantons participate to provide us with the final expression of the propagator.