I formulate a many-body theory to calculate the cross section of direct radiative neutron capture reaction by means of the Hartree-Fock-Bogoliubov mean-field model and the continuum quasiparticle random-phase approximation (QRPA). A focus is put on very-neutron-rich nuclei and low-energy neutron kinetic energy in the range from 1 keV to several MeV, which is relevant to the rapid neutron capture process of nucleosynthesis. I begin with the photoabsorption cross section and the E1 strength function. Next, in order to apply the reciprocity theorem, I decompose the cross section into partial cross sections corresponding to different channels of one-and two-neutron emission decays of photo-excited states. A numerical example is shown for the photo-absorption of Sn-142 and the neutron capture of Sn-141.