Calculable Microscopic Theory for \(^\textbf{12}\)C(\(\alpha \), \(\gamma \))\(^\textbf{16}\)O Cross Section near Gamow Window II

A microscopic approach to the \(^{12}\) $_{}^{12}$ C \((\alpha , \gamma )^{16}\) ${(\alpha ,\gamma )}^{16}$ O radiative-capture reaction near the Gamow window has been proposed by Y. Suzuki, Few-Body Syst. 62, 2 (2021). The important ingredients of the approach include the following: (1) The states of \(^{12}\) $_{}^{12}$ C and \(^{16}\) $_{}^{16}$ O relevant to the reaction are described by fully microscopic 3  \(\alpha \) $\alpha$ -particle and 4  \(\alpha \) $\alpha$ -particle configurations. (2) The isovector electric dipole transition is accounted for through the isospin impurity of the constituent \(\alpha \) $\alpha$ -particles. (3) The relative motion among the \(\alpha \) $\alpha$ -particles is expanded in terms of correlated-Gaussian basis functions. A calculation of the radiative-capture cross section demands double angular-momentum projections, that is, the angular momentum of \(^{12}\) $_{}^{12}$ C consisting of 3 \(\alpha \) $\alpha$ -particles and the orbital angular momentum for \(^{12}\) $_{}^{12}$ C \(-\alpha \) $-\alpha$ relative motion. Advancing the previous formulation based on the single angular-momentum projection, I carry out the double projection and present all the formulas needed for the cross section calculation.