Theoretical study of the direct $\alpha+d$ $\rightarrow$ $^6$Li +
  $\gamma $ astrophysical capture process in a three-body model II. Reaction
  rates and primordial abundance
  The astrophysical S-factor and reaction rate of the direct capture process
$\alpha+d$ $\rightarrow$ $^6$Li + $\gamma$, as well as the abundance of the
$^6$Li element are estimated in a three-body model. The initial state is
factorized into the deuteron bound state and the $\alpha+d$ scattering state.
The final nucleus $^6$Li(1+) is described as a three-body bound state
$\alpha+n+p$ in the hyperspherical Lagrange-mesh method. Corrections to the
asymptotics of the overlap integral in the S- and D-waves have been done for
the E2 S-factor. The isospin forbidden E1 S-factor is calculated from the
initial isosinglet states to the small isotriplet components of the final
$^6$Li(1+) bound state. It is shown that the three-body model is able to
reproduce the newest experimental data of the LUNA collaboration for the
astrophysical S-factor and the reaction rates within the experimental error
bars. The estimated $^6$Li/H abundance ratio of $(0.67 \pm 0.01)\times
10^{-14}$ is in a very good agreement with the recent measurement $(0.80 \pm
0.18)\times 10^{-14}$ of the LUNA collaboration.
