  In the present work, we use optical theorem to calculate the next-to-leading
order corrections to the QCD spectral densities directly in the QCD sum rules
for the pseudoscalar and scalar $B_c$ mesons. We take the experimental data as
guides to perform updated analysis, and obtain the masses and decay constants,
especially the decay constants, which are the fundamental input parameters in
the high energy physics, therefore the pure leptonic decay widths, which can be
confronted to the experimental data in the future.
