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

Stephen C. Anco

TL;DR

Problem: instability of a sphaleron in a one-parameter nonlinear Klein-Gordon model with a false vacuum. Approach: linear stability is transformed to Heun's equation via a $y=1/\\cosh^2 x$ change of variables, allowing eigenfunctions and eigenvalues to be expressed in terms of local Heun functions ($H\\ell$). The spectrum consists of a negative ground-state eigenvalue $\\lambda_{-1}<0$, a translation zero mode ($\\lambda_0=0$), and two positive internal modes, with a continuous branch starting at $\\lambda=4$. Key contributions: an exact Heun-function representation of the unstable mode, a continued-fraction based approximation for $\\lambda_{-1}$, and asymptotic formulas for small/large-$a$ plus explicit lump and travelling-wave solutions. Significance: provides rigorous analytic control over sphaleron instability in non-symmetric quartic potentials and clarifies the near-threshold dynamics, with implications for oscillon formation and true-vacuum expansion.

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.

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

TL;DR

Problem: instability of a sphaleron in a one-parameter nonlinear Klein-Gordon model with a false vacuum. Approach: linear stability is transformed to Heun's equation via a change of variables, allowing eigenfunctions and eigenvalues to be expressed in terms of local Heun functions (). The spectrum consists of a negative ground-state eigenvalue , a translation zero mode (), and two positive internal modes, with a continuous branch starting at . Key contributions: an exact Heun-function representation of the unstable mode, a continued-fraction based approximation for , and asymptotic formulas for small/large- plus explicit lump and travelling-wave solutions. Significance: provides rigorous analytic control over sphaleron instability in non-symmetric quartic potentials and clarifies the near-threshold dynamics, with implications for oscillon formation and true-vacuum expansion.

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 74 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Sphaleron in a false vacuum
  3. Ground states
  4. Travelling waves
  5. Linear Instability
  6. Spectrum of the perturbation potential
  7. Approximate analytical formula for the unstable mode
  8. Time scale
  9. Concluding remarks
  10. Conservation laws

Key Result

Proposition 1

Up to a shift, scaling, and reflection on $\phi$, and a dilation on $(t,x)$, any non-symmetric quartic potential with a false vacuum belongs to the $1$-parameter family The false vacuum is $V=0$ at $\phi=0$, and the true vacuum has $V<0$ at which lies between $\phi=\tanh(a)$ and $\phi={\rm coth}(a)$. The equation of motion KG.eqn in this potential is given by

Figures (9)

  • Figure 1: Potential \ref{['potential']} with a false vacuum at $\phi=0$: $a=$ 0.5, 0.8, 1.5
  • Figure 2: Lump solution \ref{['lump']} in the false-vacuum potential \ref{['potential']} for $a=$ 5.5, 2.5, 1.0, 0.5
  • Figure 3: Perturbation potential: $a=$ 0.25, ${\rm arctanh}\tfrac{1}{\sqrt{3}}$ (=0.658), 1.5
  • Figure 4: Ground state eigenvalue: $\lambda_{-1}$
  • Figure 5: $\tilde{\eta}_{-1}(y)$ for $a=$ 2.65 (space dot), 2.14 (dot), 1.44 (dash dot), 1.00 (solid), 0.881 (dash), 0.647 (space dash), 0.100 (long dash)
  • ...and 4 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Theorem 1
  • Theorem 2