Instability of sphalerons in $\phi^4$ models with a false vacuum

TL;DR

This paper investigates the instability of sphaleron solutions in a class of nonlinear Klein-Gordon models with false vacua, providing explicit eigenfunction formulations via Heun functions.

Contribution

It introduces an explicit analytical approach to study sphaleron instability in $\

Findings

Eigenfunctions and eigenvalues expressed in terms of Heun functions.

Explicit formulations enable better understanding of sphaleron stability.

Good approximations found for specific parameter ranges.

Abstract

A one-parameter family of nonlinear (quartic) Klein-Gordon models having a sphaleron solution is studied. The sphaleron arises from a saddle point between true and false vacua in the energy functional. Its instability is shown be governed by a Heun differential equation after a change of variable. This allows an explicit formulation of the eigenfunctions and eigenvalues to be obtained in terms of local Heun functions. Good approximations are found for certain ranges of the parameter.