Degenerate vacua to vacuumless model and Kbar K collisions

In this work we investigate a $Z_2$ symmetric model of one scalar field $\phi$ in $(1,1)$ dimension. The model is characterized by a continuous transition from a potential $V(\phi)$ with two vacua to the vacuumless case. The model has kink and antikink solutions that minimize energy. Stability analysis are described by a Schr\"odinger-like equation with a potential that transits from a volcano-shape with no vibrational states (in the case of vacuumless limit) to a smooth valley with one vibrational state. We are interested on the structure of 2-bounce windows present in kink-antikink scattering processes. The standard mechanism of Campbell-Schonfeld-Wingate (CSW) requires the presence of one vibrational state for the occurrence of 2-bounce windows. We report that the effect of increasing the separation of vacua from the potential $V(\phi)$ has the consequence of trading some of the first 2-bounce windows predicted by the CSW mechanism by false 2-bounce windows. Another consequence is the appearance of false 2-bounce windows of zero-order.