The $B_c$ meson and its scalar cousin with the QCD sum rules

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.