Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Three-body resonances of $ααM$ clusters ($M=φ$, $J/ψ$, $η_c$) in $^{8}_{M}{\mathrm{Be}}$ nuclei

Hao Zhou, Xiang Liu

TL;DR

This work leverages HAL QCD vector-meson nucleon potentials to model $^{8}_{M}{\mathrm Be}$ as $\alpha+\alpha+M$ three-body clusters. By folding $N$-$M$ interactions into the $\alpha$ density and solving the three-body problem with the Gaussian Expansion Method while locating resonances via Complex Scaling, it reveals a strong glue-like binding for $\phi$ that stabilizes multiple $^8$Be states and contracts the core, whereas $J/\psi$ and $\eta_c$ have weaker attractions, yielding shallow bound states or resonances and, in some cases, expanding the $\alpha$-$\alpha$ separation. A weakly bound $\alpha$-$J/\psi$ state appears in certain spin channels, arguing against a Borromean structure for $^{8}_{J/\psi}{\mathrm Be}$, while $^{8}_{\eta_c}{\mathrm Be}$ remains a candidate for Borromean behavior. The results emphasize how different vector mesons differently modulate nuclear clustering and provide concrete predictions for future experiments at facilities like J-PARC and JLab to search for such exotic hadron–nucleus systems.

Abstract

Motivated by the recently obtained HAL QCD potentials for the $N$-$φ$, $N$-$J/ψ$, and $N$-$η_c$ interactions, we investigate the structure of the exotic nuclei $^{8}_φ{\text{Be}}$, $^{8}_{J/ψ}{\text{Be}}$, and $^{8}_{η_c}{\text{Be}}$ as $α+α+M$ three-body systems ($M$ denotes the meson). The bound and resonant states are calculated consistently using the Gaussian expansion method, with resonances identified via the complex scaling method. For the $αφ$ and $α$-charmonium interactions, a folding potential is constructed based on the HAL QCD potentials and fitted to a Woods-Saxon form. We find that the $φ$ meson exhibits a strong ``glue-like" effect, binding the $0^+_1$, $2^+_1$, and $4^+_1$ resonant states of $^8$Be into stable states and significantly reducing the $α$-$α$ distance. In contrast, the interactions of $J/ψ$ and $η_c$ with the nucleus are weaker, forming only shallow bound states with the $0^+_1$ state of $^8$Be and even increasing the $α$-$α$ separation. Notably, our analysis predicts weakly bound $α$-$J/ψ$ states in the $^4S_{3/2}$ and $^2S_{1/2}$ channels, a result not reported in prior studies, which suggests that $^{8}_{J/ψ}{\text{Be}}$ may not be a Borromean nucleus. The sensitivity of the $^{8}_{M}{\mathrm{Be}}(4^+_1)$ state-transitioning from bound to resonant depending on the $α$-particle radius-highlights the subtle dynamics at play. These results provide a systematic theoretical comparison of how different vector mesons modify nuclear clustering, offering critical predictions for future experimental searches of such exotic hadron-nucleus systems.

Three-body resonances of $ααM$ clusters ($M=φ$, $J/ψ$, $η_c$) in $^{8}_{M}{\mathrm{Be}}$ nuclei

TL;DR

This work leverages HAL QCD vector-meson nucleon potentials to model as three-body clusters. By folding - interactions into the density and solving the three-body problem with the Gaussian Expansion Method while locating resonances via Complex Scaling, it reveals a strong glue-like binding for that stabilizes multiple Be states and contracts the core, whereas and have weaker attractions, yielding shallow bound states or resonances and, in some cases, expanding the - separation. A weakly bound - state appears in certain spin channels, arguing against a Borromean structure for , while remains a candidate for Borromean behavior. The results emphasize how different vector mesons differently modulate nuclear clustering and provide concrete predictions for future experiments at facilities like J-PARC and JLab to search for such exotic hadron–nucleus systems.

Abstract

Motivated by the recently obtained HAL QCD potentials for the -, -, and - interactions, we investigate the structure of the exotic nuclei , , and as three-body systems ( denotes the meson). The bound and resonant states are calculated consistently using the Gaussian expansion method, with resonances identified via the complex scaling method. For the and -charmonium interactions, a folding potential is constructed based on the HAL QCD potentials and fitted to a Woods-Saxon form. We find that the meson exhibits a strong ``glue-like" effect, binding the , , and resonant states of Be into stable states and significantly reducing the - distance. In contrast, the interactions of and with the nucleus are weaker, forming only shallow bound states with the state of Be and even increasing the - separation. Notably, our analysis predicts weakly bound - states in the and channels, a result not reported in prior studies, which suggests that may not be a Borromean nucleus. The sensitivity of the state-transitioning from bound to resonant depending on the -particle radius-highlights the subtle dynamics at play. These results provide a systematic theoretical comparison of how different vector mesons modify nuclear clustering, offering critical predictions for future experimental searches of such exotic hadron-nucleus systems.

Paper Structure

This paper contains 8 sections, 30 equations, 6 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. Three-body resonances of $\alpha\alpha M$ cluster
  3. Hamiltonian
  4. Gaussian Expansion Method
  5. Complex scaling method
  6. numerical results
  7. The $\prescript{8}{J/\psi}{\text{Be}}$ and $\prescript{8}{\eta_c}{\text{Be}}$ systems
  8. SUMMARY

Figures (6)

  • Figure 1: Recoupling of $[[s_3s_4]_0 s_5]_{1m}$ into a superposition of $[s_3[s_4s_5]_{1/2}]_{1m}$ and $[s_3[s_4s_5]_{3/2}]_{1m}$. The coefficients $-\sqrt{1/3}$ and $\sqrt{2/3}$ yield the statistical weights $1/3$ and $2/3$ used in the spin‑averaged potential.
  • Figure 2: The $V_{\alpha\phi}(r)$/$V_{\alpha J/\psi}(r)$ potentials corresponding to the $N$-$\phi$/$J/\psi$ interaction in the ${}^4S_{3/2}$ channel when the $\alpha$-particle rms matter radius $R_\alpha=1.84$, 1.70, and 1.56 fm/$1.70\sqrt{1.5}$, 1.70, and 1.47 fm.
  • Figure 3: Three Jacobian coordinates of three-body system.
  • Figure 4: The energy spectra of $\prescript{8}{}{\text{Be}}$ from phenomenological $\alpha$-$\alpha$ potential and $\prescript{9}{\Lambda}{\text{Be}}$ from Ref. Lee:2019mltWu:2019ivs, as well as the energy spectra of $\prescript{8}{\phi}{\text{Be}}$, $\prescript{8}{J/\psi}{\text{Be}}$ and $\prescript{8}{\eta_c}{\text{Be}}$ with an $\alpha$-particle rms matter radius of 1.70 fm. The values in parenthesis are decay widths, while the dashed lines represent the decay thresholds. Note that the spacing between $y$-axis $-$20 and $-$15 has been compressed for aesthetic purposes. The content in parentheses after $\prescript{8}{M}{\text{Be}}$ refers to the $\alpha$-$M$ interaction used.
  • Figure 5: Dependence of the complex energy eigenvalues on the scaling angle $\theta$ for $\prescript{8}{\phi}{\text{Be}}(4_1^+)$ using the $V_{\alpha\phi}^{3/2}(r)$ potential with an $\alpha$-particle rms matter radius of 1.84 fm.
  • ...and 1 more figures