Proton-halo breakup dynamics for the breakup threshold in the ε0 → 0 limit

Proton-halo breakup behavior in the \begin{document}$ \varepsilon_0\to 0 $\end{document} limit (where \begin{document}$ \varepsilon_0 $\end{document} is the ground-state binding energy) is studied around the Coulomb barrier in the \begin{document}$ ^8{\rm{B}}+{}^{58}{\rm{Ni}} $\end{document} reaction for the first time. For practical purposes, apart from the experimental \begin{document}$ ^8{\rm{B}} $\end{document} binding energy of 137 keV, three more arbitrarily chosen values (1, 0.1, 0.01 keV) are considered. It is first shown that the Coulomb barrier between the core and the proton prevents the \begin{document}$ ^7{\rm{Be}}+p $\end{document} system from reaching the state of an open proton-halo system, which, among other factors, would require the ground-state wave function to extend to infinity in the asymptotic region, as \begin{document}$ \varepsilon_0\to 0 $\end{document}. The elastic scattering cross section, which depends on the density of the ground-state wave function, is found to have a negligible dependence on the binding energy in this limit. The total, Coulomb and nuclear breakup cross sections are all reported to increase significantly from \begin{document}$ \varepsilon_0 = 137 $\end{document} to 1.0 keV, and converge to their maximum values as \begin{document}$ \varepsilon_0\to 0 $\end{document}. This increase is mainly understood as coming from a longer tail of the ground-state wave function for \begin{document}$ \varepsilon_0\leqslant 1.0 $\end{document} keV, compared to that for \begin{document}$ \varepsilon_0 = 137 $\end{document} keV. It is also found that the effect of the continuum-continuum couplings is to slightly delay the convergence of the breakup cross section. The analysis of the reaction cross section indicates a convergence of all the breakup observables as \begin{document}$ \varepsilon_0\to 0 $\end{document}. These results provide a better sense of the dependence of the breakup process on the breakup threshold.