Primordial ${\alpha} + d \to {}^{6}{\rm Li} + \gamma$ reaction and second Lithium puzzle

During the Big Bang, ${}^{6}{\rm Li}$ was synthesized via the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction. After almost 25 years of the failed attempts to measure the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction in the lab at the Big Bang energies, just recently the LUNA collaboration presented the first successful measurements at two different Big Bang energies [M. Anders {\it et al.}, Phys. Rev. Lett. {\bf 113}, 042501 (2014)]. In this paper we will discuss how to improve the accuracy of the direct experiment. To this end the photon's angular distribution is calculated in the potential model. It contains contributions from electric dipole and quadrupole transitions and their interference, which dramatically changes the photon's angular distribution. The calculated distributions at different Big Bang energies have a single peak at $\sim 50^{\circ}$. These calculations provide the best kinematic conditions to measure the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction. The expressions for the total cross section and astrophysical factor are also derived by integrating the differential cross section over the photon's solid angle. The LUNA data are in excellent agreement with our calculations using a potential approach combined with a well established asymptotic normalization coefficient for ${}^{6}{\rm Li} \to \alpha +d$. Comparisons of the available experimental data for the $S_{24}$ astrophysical factor and different calculations are presented. The Big Bang lithium isotopic ratio ${}^{6}{\rm Li}/^{7}{\rm Li} = (1.5 \pm 0.3)\times 10^{-5}$ following from the LUNA data and the present analysis are discussed in the context of the disagreement between the observational data and the standard Big Bang model, which constitutes the second Lithium problem.