  The $\alpha+d\rightarrow\, ^6{\rm Li}+\gamma$ radiative capture is studied in
order to predict the $^6$Li primordial abundance. Within a two-body framework,
the $\alpha$ particle and the deuteron are considered the structureless
constituents of $^6$Li. Five $\alpha+d$ potentials are used to solve the
two-body problem: four of them are taken from the literature, only one having
also a tensor component. A fifth model is here constructed in order to
reproduce, besides the $^6$Li static properties as binding energy, magnetic
dipole and electric quadrupole moments, also the $S$-state asymptotic
normalization coefficient (ANC). The two-body bound and scattering problem is
solved with different techniques, in order to minimize the numerical
uncertainty of the present results. The long-wavelength approximation is used,
and therefore only the electric dipole and quadrupole operators are retained.
The astrophysical $S$-factor is found to be significantly sensitive to the ANC,
but in all the cases in good agreement with the available experimental data.
The theoretical uncertainty has been estimated of the order of few % when the
potentials which reproduce the ANC are considered, but increases up to $\simeq
20$ % when all the five potential models are retained. The effect of this
$S$-factor prediction on the $^6$Li primordial abundance is studied, using the
public code PArthENoPE. For the five models considered here we find $^6{\rm
Li}/$H$ = (0.9 - 1.8) \times 10^{-14}$, with the baryon density parameter in
the 3-$\sigma$ range of Planck 2015 analysis, $\Omega_b h^2= 0.02226 \pm
0.00023$.
