Fake Instability in the Euclidean Formalism

TL;DR

This paper revises the Euclidean path-integral formalism for metastable states, introducing valley instantons to correctly compute energy spectra without spurious imaginary parts, thus extending the method's validity.

Contribution

It introduces valley instantons and a global path-space analysis to improve the Euclidean formalism for metastable states, avoiding false imaginary energy contributions.

Findings

Proper valley instantons dominate the path integral.

The energy spectrum can be obtained without imaginary parts.

The method extends the Euclidean formalism to global path-space considerations.

Abstract

We study the path-integral formalism in the imaginary-time to show its validity in a case with a metastable ground state. The well-known method based on the bounce solution leads to the imaginary part of the energy even for a state that is only metastable and has a simple oscillating behavior instead of decaying. Although this has been argued to be the failure of the Euclidean formalism, we show that proper account of the global structure of the path-space leads to a valid expression for the energy spectrum, without the imaginary part. For this purpose we use the proper valley method to find a new type of instanton-like configuration, the ``valley instantons''. Although valley instantons are not the solutions of equation of motion, they have dominant contribution to the functional integration. A dilute-gas approximation for the valley instantons is shown to lead to the energy formula. This method extends the well-known imaginary-time formalism so that it can take into account the global behavior of the theory.