Gamow shell model description of the radiative capture reaction B8(p,γ)C9

Background: In low metallicity supermassive stars, the hot $pp$ chain can serve as an alternative to produce the CNO nuclei. In the astrophysical environment of high temperature, the proton capture of $^{8}\mathrm{B}$ can be faster than its $\ensuremath{\beta}$ decay, so that the $^{8}\mathrm{B}(p,\ensuremath{\gamma})^{9}\mathrm{C}$ reaction plays an important role in the hot $pp$ chain. Due to the unstable nature of $^{8}\mathrm{B}$ and the unavailability of $^{8}\mathrm{B}$ beams, the study of the $^{8}\mathrm{B}(p,\ensuremath{\gamma})^{9}\mathrm{C}$ reaction can only be achieved by indirect methods, so that large uncertainties exist.Purpose: The Gamow shell model in the coupled-channel representation (GSM-CC) is applied to study the proton radiative capture reaction $^{8}\mathrm{B}(p,\ensuremath{\gamma})^{9}\mathrm{C}$.Method: The GSM-CC is a unified microscopic theory for the description of nuclear structure and nuclear reaction properties. A translationally invariant Hamiltonian is considered for that matter, making use of a finite-range two-body interaction, whose parameters are adjusted to reproduce the low-energy spectra of $^{8}\mathrm{B}$ and $^{9}\mathrm{C}$. The reaction channels are then built through the coupling of the wave functions of the ground state ${2}_{1}^{+}$, the first excited state ${1}_{1}^{+}$, and the second excited state ${3}_{1}^{+}$ in $^{8}\mathrm{B}$ to a projectile proton wave function in different partial waves. For the calculation of the $^{8}\mathrm{B}(p,\ensuremath{\gamma})^{9}\mathrm{C}$ astrophysical factor, all $E1, M1$, and $E2$ transitions from the initial continuum states to the final bound states ${3/2}_{1}^{\ensuremath{-}}\mathrm{of}\phantom{\rule{4pt}{0ex}}^{9}\mathrm{C}$ are considered. The resonant capture to the first resonant state ${1/2}_{1}^{\ensuremath{-}}$ of $^{9}\mathrm{C}$ is also calculated.Results: The experimental low-energy levels and the proton emission threshold in $^{9}\mathrm{C}$ are reproduced by the GSM-CC. The calculated astrophysical factor agrees with experimental data obtained from indirect measurements. The reaction rates from the direct capture and resonant capture are calculated for the temperature range of astrophysical interest.Conclusion: The calculated total astrophysical $S$ factor is dominated by the $E1$ transition to the ground state of $^{9}\mathrm{C}$. The GSM-CC calculations suggest that $S$ first increases with the energy of the center of mass ${E}_{\mathrm{c}.\mathrm{m}.}$, and then decrease with the energy. This agrees with existing data, which has smaller values around zero energy and larger values in the energy range of 0.2 MeV $\ensuremath{\le}{E}_{\mathrm{c}.\mathrm{m}.}\ensuremath{\le}$ 0.6 MeV.