Elastic Kink-Meson Scattering in the ${\mathrm{\Phi }}^4$ Double-Well Model

We calculate the leading order amplitude and probability for the elastic scattering of an elementary meson and a kink in the ${\phi }^4$ 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.