(De-)Exciting the Third Poschl-Teller Kink

There is a series of scalar models possessing reflectionless kinks whose linear perturbations are described by a P\"oschl-Teller potential at integer level $\sigma$. The cases $\sigma=1$ and $2$ are the well-known Sine-Gordon and $\phi^4$ double-well models. The $\sigma=3$ kink has received relatively little attention because it exhibits a $\phi^{8/3}$ 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 $\sigma=3$ model is not pathological, but rather its mesons are quantum field theoretic extensions of Znojil's bound states.