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

A microscopic approach to the $^{12}$C$(\alpha, \gamma)^{16}$O radiative-capture reaction near the Gamow window has been proposed by Y. Suzuki, Few-Body Syst. {\bf 62}, 2 (2021). The important ingredients of the approach include the following: (1) The states of $^{12}$C and $^{16}$O relevant to the reaction are described by fully microscopic 3\,$\alpha$-particle and 4\,$\alpha$-particle configurations. (2) The isovector electric dipole transition is accounted for through the isospin impurity of the constituent $\alpha$-particles. (3) The relative motion among the $\alpha$-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 $^{12}$C consisting of 3 $\alpha$-particles and the orbital angular momentum for $^{12}$C$-\alpha$ 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.