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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 isfactorized 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 theasymptotics of the overlap integral in the S- and D-waves have been done forthe E2 S-factor. The isospin forbidden E1 S-factor is calculated from theinitial 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 toreproduce the newest experimental data of the LUNA collaboration for theastrophysical S-factor and the reaction rates within the experimental errorbars. The estimated $^6$Li/H abundance ratio of $(0.67 \pm 0.01)\times10^{-14}$ is in a very good agreement with the recent measurement $(0.80 \pm0.18)\times 10^{-14}$ of the LUNA collaboration.

physical review. c

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