Elastic Kink-Meson Scattering in the $Φ^4$ Double-Well Model

TL;DR

The paper computes the leading-order elastic scattering amplitude for a meson off a φ^4 kink in (1+1)D, a non‑integrable soliton–meson system.It uses the quantum displacement operator framework within Linearized Soliton Perturbation Theory to perform a mode‑decomposed, one‑loop calculation around the kink background.A pole appears in the amplitude when the incoming meson energy matches twice the shape‑mode energy, signaling an unstable resonance that should acquire a width from higher‑order corrections.Compared with the integrable sine‑Gordon case where elastic scattering vanishes, the φ^4 model exhibits a finite, energy‑dependent elastic amplitude with a resonance and continuum thresholds, outlining a framework for resonances in non‑integrable soliton dynamics.

Abstract

We calculate the leading order amplitude and probability for the elastic scattering of an elementary meson and a kink in the $φ^4$ double-well model. Classically, the kink is reflectionless, and so the leading contribution arises at one loop. At this order, the scattering amplitude exhibits a pole when the incoming meson energy is twice the shape mode energy, corresponding to the excitation of an unstable resonance with the twice excited shape mode. We expect that higher order corrections will give this resonance a width equal to the inverse of the known lifetime of this unstable excitation.

Figures (2)

• Figure 1: The diagrams represent the contributions of individual amplitudes to the elastic scattering amplitude $R(k_0)$. Note that here time starts from the right.
• Figure 2: Contributions $A(k_0)$, $B(k_0)$, $C(k_0)$ and $D(k_0)$ to the elastic scattering in the $\phi^4$ model