  We calculate the radiative capture cross section for $^7$Be($\alpha,
\gamma$)$^{11}$C and its reaction rate of relevance for the big bang
nucleosynthesis. The impact of this reaction on the primordial $^7$Li abundance
is revised including narrow and broad resonances in the pertinent energy
region. Our calculations show that it is unlikely that very low energy
resonances in $^{11}$C of relevance for the big bang nucleosynthesis would
emerge within a two-body potential model. Based on our results and a comparison
with previous theoretical and experimental analyses, we conclude that the
impact of this reaction on the so-called "cosmological lithium puzzle" is
completely irrelevant.
