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(De-)Exciting the Third Poschl-Teller Kink

Hengyuan Guo, Jarah Evslin, Stefano Bolognesi

Abstract

There is a series of scalar models possessing reflectionless kinks whose linear perturbations are described by a Pöschl-Teller potential at integer level $σ$. The cases $σ=1$ and $2$ are the well-known Sine-Gordon and $φ^4$ double-well models. The $σ=3$ kink has received relatively little attention because it exhibits a $φ^{8/3}$ potential, whose third derivative diverges in the vacuum. In old-fashioned perturbation theory this yields a cubic interaction that diverges far from a kink. We nonetheless use this interaction to calculate the amplitudes and probabilities for incoming radiation to excite or de-excite one of the kink's two shape modes. As each shape mode is localized about the kink, the leading order amplitudes are nonetheless finite. This suggests that the $σ=3$ model is not pathological, but rather its mesons are quantum field theoretic extensions of Znojil's bound states.

(De-)Exciting the Third Poschl-Teller Kink

Abstract

There is a series of scalar models possessing reflectionless kinks whose linear perturbations are described by a Pöschl-Teller potential at integer level . The cases and are the well-known Sine-Gordon and double-well models. The kink has received relatively little attention because it exhibits a potential, whose third derivative diverges in the vacuum. In old-fashioned perturbation theory this yields a cubic interaction that diverges far from a kink. We nonetheless use this interaction to calculate the amplitudes and probabilities for incoming radiation to excite or de-excite one of the kink's two shape modes. As each shape mode is localized about the kink, the leading order amplitudes are nonetheless finite. This suggests that the model is not pathological, but rather its mesons are quantum field theoretic extensions of Znojil's bound states.
Paper Structure (17 sections, 59 equations, 4 figures)

This paper contains 17 sections, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Methods
  4. Classical Models Yielding Pöschl-Teller Kinks
  5. General Models
  6. Pöschl-Teller Potentials
  7. Soliton with general PT potential
  8. Stokes and Anti-Stokes Scattering
  9. General Formulas
  10. Finiteness
  11. The $\sigma=3$ Kink
  12. Scattering Probabilities
  13. Expanding the Potential about its Minimum
  14. The Center of the Kink
  15. The Asymptotic Vacua
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Left to right: $V"'[f(x)]$ at $m=1$ for different values of $\sigma$.
  • Figure 2: Two shape modes and one zero mode in the $\sigma=3$ PT potential
  • Figure 3: Stokes (left) and anti-Stokes (right) scattering probabilities as functions of $k_0$ for the first shape mode in the case $\sigma=3$. The bottom panels are zoomed in on energies just above their respective thresholds.
  • Figure 4: Stokes (left) and anti-Stokes (right) scattering probabilities as functions of $k_0$ for the second shape mode in the case $\sigma=3$. The bottom panels are zoomed in on energies just above their respective thresholds.