Structure of the $^8$B and $^8$Li nuclei and the astrophysical $S_{17}(0)$-factor of the $^7$Be($p,γ$)$^8$B direct capture process within a three-body model

Abstract

The structure of the ground $(2^+)$ and excited $(1^+)$ bound states of the $^8$B and $^8$Li nuclei is studied within the framework of the $α+^3$He($^3$H)+$p(n)$ three-body potential cluster model based on the hyperspherical Lagrange-mesh method. The two-body realistic potentials have been applied from the literature. Convergent theoretical estimates for the three-body binding energy and matter radius have been obtained with the maximal hypermomentum $K_{max}=22$ and 28 for the ground and excited $1^+$ states, respectively. The ANC value of the virtual transition of the $^8$B nucleus is estimated self-consistently by matching the overlap integral of the $^8$B three-body and the $^7$Be two-body wave functions with it's asymptotics. The obtained values are $0.211$~fm$^{-1/2}$ and $0.739$~fm$^{-1/2}$ in the spin 1 and spin 2 channels, respectively. For the ANC values of the $^8$Li nucleus the estimates $0.220$~fm$^{-1/2}$ and $0.774$~fm$^{-1/2}$ are extracted. The ratio $C^2(^8 {\rm B})/C^2(^8 {\rm Li})=0.912$ implies a breaking of the mirror symmetry of the strong nuclear forces of order 27\% due to the Coulomb interaction and the dynamical three-body effects. For the $S_{17}(0)$ -factor an estimate $22.492\pm0.014$ eV b was obtained based on the asymptotic theory developed by D. Baye [Phys. Rev. C {\bf 62},065803 (2000)]. The spin 2 channel contributes with $S^{(2)}_{17}(0)=20.838 \pm 0.014$ eV b, while the spin 1 channel yields $S^{(1)}_{17}(0)=1.654 \pm 0.003$ eV b. These results for $S_{17}(0)$ are in a good agreement with the estimate $20.8\pm0.7{\rm(th)}\pm1.4{\rm(exp)}$ eV b of the SF II, but larger than the recommended value $20.5\pm0.70$ eV b of the SF III. At the same time, our estimate is very close to the value 22.4 eV b used in the most successful Solar Model BAR2M [W.~Yang and Z.~Tian, AJ {\bf 970} (2024), 38].