Critical properties of bound states with one-boson-exchange potential

TL;DR

This work addresses the global, critical properties of bound states in one-boson-exchange potentials by deriving a Schrödinger equation from the Bethe-Salpeter framework under the instantaneous approximation, yielding Yukawa-type potentials $V_0(r)$ and form-factor–modified $V_1(r)$. A high-precision shooting method is developed to compute ground- and excited-state thresholds and the corresponding critical screening masses, with results for the ground state $m_{ex}$ in agreement with literature to ~30 significant digits and substantially more precise data for higher angular momenta. The study reveals that, while form factors modify the detailed dependence on the cutoff $\Lambda$, the number of bound states is largely governed by the coupling $\alpha$ and exchange masses, a global regularity that extends to hadronic molecules such as the $D\bar{D}$ system where the ground-state critical coupling is nearly cutoff-independent for large $\Lambda$. These findings provide a robust, high-accuracy framework for assessing hadronic molecular states and can be extended to higher $l$ or other systems with minimal modification.

Abstract

In this study, we discuss some general critical properties of bound states with one-boson-exchange potential. For simplicity, we first take a system with two identical scalar particles as an example. The interaction between these two scalar particles is described by the exchange of another massive scalar meson under the instantaneous approximation, which results in the Yukawa potential. A highly accurate numerical method is used to determine the critical screening mass value of the system. The resulting critical screening mass for the ground state is consistent with those reported in the literature, agreeing to about 30 significant figures. The highly accurate results for the $l=1$ case are also presented, which are significantly more precise than those previously reported in the literature. Furthermore, we extend the discussion to physical hadronic molecule states, where form factors are introduced in the interaction to describe the structure of hadrons. Our numerical results show that although the binding energies of the hadronic molecule states depend on the cutoff in the form factors, the number of hadronic molecule states is almost independent of the cutoffs across a very wide physical region. This indicates a strong and important property: the number of hadronic molecule states is almost solely determined by the coupling constants and the masses of the exchange particles. This highly accurate numerical method can also be straightforwardly applied to higher $l$ cases or other systems.

Figures (6)

• Figure 1: The number of bound states on the parameters $m_{ex}$ and $\alpha$ for specifical angular momentum $l$, where $\mu$ is fixed as $1$. The panels, ordered from left to right, illustrate the cases for $l=0$, $l=1$, and $l=2$. The red region (I), blue region (II), and green region (III) areas signify the existence of one, two, and three bound states, respectively.
• Figure 2: The number of bound states dependence on the parameters $m_{ex}$ and $\alpha$ for specific angular momentum $l$ and cut-off $\Lambda$, where $\mu$ is fixed as $1$. The panels, ordered from left to right, illustrate the cases for $l=0$, $l=1$, and $l=2$. The red (I), blue (II), and green (III) regions signify the existence of one, two, and three bound states, respectively.
• Figure 3: The number of bound states dependence on the parameters $\Lambda$ and $\alpha$ for specific angular momentum $l$ and cut-off $\Lambda$, where $\mu$ is fixed as $1$ and $m_{ex}$ is fixed as $0.1$. The panels, ordered from left to right, illustrate the cases for $l=0$, $l=1$, and $l=2$. The red (I), blue (II), and green (III) regions signify the existence of one, two, and three bound states, respectively.
• Figure 4: Dependence of the number of bound states for the $D\bar{D}$ system on the parameters $\Lambda$ and $\alpha$ for specific angular momentum $l$ and cut-off $\Lambda$. The panels, ordered from left to right, illustrate the cases for $l=0$, $l=1$, and $l=2$. The red (I), blue (II), and green (III) regions signify the existence of one, two, and three bound states, respectively.
• Figure 5: Dependence of $E_b$ on $\alpha$ with fixed $\Lambda$ in the $l=0$ case. The panels, ordered from left to right, illustrate the cases for $\Lambda=1.46$ GeV, $\Lambda=1.76$ GeV, and $\Lambda=5$ GeV. The red, blue, and green lines signify the first, second, and third bound states, respectively. The two black triangles correspond to results using the parameters given in Ref. Wang:2021aql.