Calculable microscopic theory for $^{12}$C($\alpha, \gamma$)$^{16}$O cross section near Gamow window

$^{12}$C$(\alpha, \gamma)^{16}$O radiative-capture process is a key reaction to produce the element of oxygen in stars. Measuring the cross section near the Gamow window is extremely hard because it is too small. To make a theoretical contribution towards resolving the long-standing problem, I present a microscopic formulation that aims at providing all materials needed to calculate the cross section. The states of $^{12}$C and $^{16}$O relevant to the reaction are respectively described with fully microscopic 3 $\alpha$-particle and 4 $\alpha$-particle configurations, in which the relative motion among the $\alpha$ particles is expanded in terms of correlated Gaussian basis functions. The configuration space has the advantage of being able to well describe the reduced $\alpha$-width amplitudes of the states of $^{16}$O. Both electric dipole and electric quadrupole transitions are responsible for the radiative-capture process. The $\alpha$ particle is described with a $(0s)^4$ configuration admixed with a small amount of an isospin $T=1$ impurity component, which is crucially important to account for the isovector electric dipole transition. The isoscalar electric dipole operators are also taken into account up to the first order beyond the long-wavelength approximation. All the necessary ingredients are provided to make the paper self-contained and ready for numerical computations.