Inverse scattering problem with a bare state

Abstract

In hadron physics, molecular-like multihadron states can interact with compact multiquark states. The latter are modeled as bare states in the Hilbert space of a potential model. In this work, we study several potential models relevant to the bare state, and solve their inverse scattering problems. The first model, called ``cc,'' is a separable potential model. We show that it can approximate (single-channel short-range) $S$-wave near-threshold physics with an error of $\mathcal{O}({\ensuremath{\beta}}^{3}/{M}_{V}^{3})$, where $\ensuremath{\beta}$ sets the maximum momentum of the near-threshold region and ${M}_{V}$ is the typical scale of the potential. The second model, called ``bc,'' serves as the bare-state-dominance approximation, where interaction between continuum states is ignored. Under this model, even though the bare state is always crucial for a bound state's generation, a shallow bound state naturally tends to have a small bare-state proportion. Therefore, we need other quantities to quantify the importance of the bare state. The last model, called ``bcc,'' is a combination of the first two models. This model not only serves as a correction to the bare-state-dominance approximation, but can also be used to understand the interplay between quark and hadron degrees of freedom. This model naturally leads to the presence of a Castillejo-Dalitz-Dyson (CDD) zero. We consider the energy decomposition of a bound state. The potential ratio of the bare-continuum interaction to the continuum self-interaction is proposed to understand how the bound state is generated. Model independently, an inequality for the potential ratio is derived. Based on the model ``bcc,'' the CDD zero can be used to estimate the potential ratio. Finally, we apply these studies to the deuteron, $\ensuremath{\rho}$ meson, and ${D}_{s0}^{*}(2317)$, and analyze their properties.