Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long-time behaviour of sphalerons in $φ^4$ models with a false vacuum

Stephen C. Anco, Danial Saadatmand

TL;DR

This work analyzes the long-time dynamics of sphalerons in a 1+1 dimensional nonlinear Klein-Gordon model with a false vacuum. It combines numerical simulations perturbing the sphaleron along its unstable mode with a nonlinear collective-coordinate reduction, yielding a power-series analytical solution that describes how the sphaleron evolves into an accelerating kink-antikink pair whose flanks asymptotically approach the light cone. The results show energy concentrating at the moving flanks and a gradient blow-up in the large-time limit, confirming the analytical picture and aligning with numerical observations. The findings illuminate energy localization mechanisms in nonlinear field theories and lay the groundwork for future studies of oscillon channels and sphaleron–sphaleron collisions.

Abstract

Sphalerons in nonlinear Klein-Gordon models are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation approaches the speed of light asymptotically at large times. Utilizing a nonlinear collective coordinate method, an approximate analytical solution is derived for this evolution. These results indicate that an exact solution is expected to exhibit a gradient blow-up for large times,caused by energy concentrating at the flanks of the kink-antikink pair.

Long-time behaviour of sphalerons in $φ^4$ models with a false vacuum

TL;DR

This work analyzes the long-time dynamics of sphalerons in a 1+1 dimensional nonlinear Klein-Gordon model with a false vacuum. It combines numerical simulations perturbing the sphaleron along its unstable mode with a nonlinear collective-coordinate reduction, yielding a power-series analytical solution that describes how the sphaleron evolves into an accelerating kink-antikink pair whose flanks asymptotically approach the light cone. The results show energy concentrating at the moving flanks and a gradient blow-up in the large-time limit, confirming the analytical picture and aligning with numerical observations. The findings illuminate energy localization mechanisms in nonlinear field theories and lay the groundwork for future studies of oscillon channels and sphaleron–sphaleron collisions.

Abstract

Sphalerons in nonlinear Klein-Gordon models are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation approaches the speed of light asymptotically at large times. Utilizing a nonlinear collective coordinate method, an approximate analytical solution is derived for this evolution. These results indicate that an exact solution is expected to exhibit a gradient blow-up for large times,caused by energy concentrating at the flanks of the kink-antikink pair.

Paper Structure

This paper contains 21 sections, 1 theorem, 95 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Sphaleron in a false vacuum
  3. Sphalerons
  4. Linear Instability
  5. Numerical solution
  6. Numerical solutions with initial growth
  7. Collective coordinates for sphalerons
  8. Series solution
  9. Asymptotic behaviour
  10. Approximate behaviour of the perturbed sphaleron
  11. Perturbation of the sphaleron at early times
  12. Example $b=0.998$ ($a=2.0$)
  13. Example $b=0.95$ ($a=1.174$)
  14. Example $b=0.7$ ($a=0.616$)
  15. Example $b=0.5$ ($a=0.402$)
  16. ...and 6 more sections

Key Result

Theorem 1

The variational equations ansatz.eom have a lump solution given by an explicit power series involving two free parameters $T$ and $B_0$. The conserved energy of the solution is given by

Figures (24)

  • Figure 1: Potential with a false vacuum at $\phi=0$: $a=$ 0.5, 0.8, 1.5
  • Figure 2: Perturbation potential: $a=$ 0.25, ${\rm arctanh}\tfrac{1}{\sqrt{3}}$ (=0.658), 1.5
  • Figure 3: Ground-state eigenfunction, normalized in $L^2$, for $a=$ 2.65, 2.14, 1.44, 1.00, 0.881, 0.647, 0.100
  • Figure 4: Ground-state eigenvalue: $\lambda_{-1}$
  • Figure 5: Numerical solution for $a=1.174$. (Left) Short times $t=0,2,4,5$; (Right) Long times $t=10,30,40,80$.
  • ...and 19 more figures

Theorems & Definitions (1)

  • Theorem 1