Piecewise linear potentials for false vacuum decay and negative modes

TL;DR

This paper analytically investigates bounce solutions and negative modes in piecewise linear potentials, providing exact solutions and confirming the presence of a single negative mode, thus supporting their use in false vacuum decay studies.

Contribution

It introduces an analytical approach to bounce solutions and negative modes in piecewise linear potentials, simplifying calculations in false vacuum decay analysis.

Findings

Bounce solutions and actions are obtained analytically in any spacetime dimension.

The Schrödinger-like fluctuation equations are solved exactly for negative modes.

There is exactly one negative mode in these potentials.

Abstract

We study bounce solutions and associated negative modes in the class of piecewise linear triangular-shaped potentials that may be viewed as approximations of smooth potentials. In these simple potentials, the bounce solution and action can be obtained analytically for a general spacetime dimension $D$. The eigenequations for the fluctuations around the bounce are universal and have the form of a Schr\"odinger-like equation with delta-function potentials. This Schr\"odinger equation is solved exactly for the negative modes whose number is confirmed to be one. The latter result may justify the usefulness of such piecewise linear potentials in the study of false vacuum decay.