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

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