Negative Mode Problem of False Vacuum Decay Revisited

TL;DR

This paper investigates the negative mode problem in false vacuum decay, comparing Lagrangian and Hamiltonian formulations, and clarifies how variable choices influence negative mode conditions.

Contribution

It clarifies the discrepancy between Lagrangian and Hamiltonian approaches regarding negative modes and identifies variable choices that unify the conditions.

Findings

Different formulations have inconsistent negative mode conditions.

Variable choice impacts the negative mode analysis.

A specific variable choice aligns Hamiltonian results with Lagrangian ones.

Abstract

We study negative modes around the Coleman-de Luccia bounce solution. The conditions for the appearance of an infinite number of the negative modes do not coincide between Lagrangian and Hamiltonian formulations in the literature, and we discuss the origin of this difference in detail. We show how different choices of the variable for the fluctuation around the bounce solution affect the negative mode condition in the Hamiltonian approach, and point out that there exists a choice which gives the same negative mode condition as the Lagrangian approach.