Structure of Bound States with Coulomb plus Short-range Interaction

TL;DR

The paper investigates how the Coulomb interaction modifies near-threshold bound states formed by short-range forces in a nonrelativistic two-body system. It solves a Coulomb-plus-square-well potential $V(r)=V_{0}\Theta(b-r)+\frac{\alpha Z_{1}Z_{2}}{r}$ for $s$-wave bound states, analyzing binding energies $B$, radial wave functions $u(r)$, and observables such as $\langle r^2\rangle$. The results show that Coulomb shifts depend on the spatial structure of the bound state, with repulsion reducing $B$ and attraction increasing it; near-threshold scaling changes from the short-range universal $B\propto V_{0}^2$ to a linear $B\propto V_{0}$ in the presence of Coulomb, and the weakly bound state remains localized with a non-unity compositeness as $B\to0$. These findings have implications for interpreting near-threshold exotic hadrons and demonstrate that long-range Coulomb forces can qualitatively modify universal low-energy behavior and bound-to-resonance transitions, especially in systems where Coulomb attraction or repulsion competes with short-range dynamics.

Abstract

We study the structure of bound states appearing in systems governed by the Coulomb and short-range interactions. We analyze the binding energies and wave functions of the bound states generated by the Coulomb plus short-range potential. We demonstrate that Coulomb-induced shifts of the binding energy are closely correlated with the spatial distribution of the wave function. Furthermore, we show that the asymptotic behavior of wave functions of weakly bound states is qualitatively altered by Coulomb repulsion, leading to a modification of the near-threshold mass scaling that is otherwise universal for short-range interactions.

Figures (3)

• Figure 1: Radial density distributions of bound states, $|u(r)|^{2}$, obtained with the square-well potential. Left (right) panels show the ground (excited) state. Top (bottom) panels correspond to $V_{0}=-125$ MeV ($-250$ MeV). The vertical dashed lines indicate the interaction range $r=b$.
• Figure 2: Dependence of the binding energy $B$ on the well depth $-V_{0}$ for $Z_{1}Z_{2}=0$ (dotted line), $+1$ (dashed line), and $+4$ (solid line).
• Figure 3: Density distributions of weakly bound states with binding energies $B = 2.0$, $1.0$, and $0.05$ MeV. Left: $Z_{1}Z_{2}=0$. Right: $Z_{1}Z_{2}=+4$.