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The astrophysical $S$ factor and reaction rates of the direct capture process$^{6}$Li(p,$\gamma)^{7}$Be are estimated within a two-body single-channelpotential model approach. Central potentials of the Gaussian-form in the$^2P_{3/2}$ and $^2P_{1/2}$ waves are adjusted to reproduce the bindingenergies and the empirical values of the asymptotic normalization coefficients(ANC) for the $^7$Be(3/2$^-$) ground and $^7$Be(1/2$^-$) excited bound states,respectively. The parameters of the potential in the most important $^2S_{1/2}$scattering channel were fitted to reproduce the empirical phase shifts from theliterature and the low-energy astrophysical $S$ factor of the LUNAcollaboration. The obtained results for the astrophysical $S$ factor and thereaction rates are in a very good agreement with available experimental datasets. The numerical estimates reproduce not only the absolute values, but alsothe energy and temperature dependence of the $S$ factor and reaction rates ofthe LUNA collaboration, respectively. The estimated $^{7}{\rm Li/H}$ primordialabundance ratio $(4.67\pm 0.04 )\times 10^{-10}$ is well consistent with recentBBN result of $(4.72\pm 0.72) \times 10^{-10}$ after the Planck observation.

Detailed study of the astrophysical direct capture reaction $^{6}{\rm Li}(p, \gamma)^{7}{\rm Be}$ in a potential model approach