Two-loop QCD Corrections to $B_c$ Meson Leptonic Decays

The two-loop QCD radiative corrections to the $B_c$ meson leptonic decay rate are calculated in the framework of NRQCD factorization formalism. Two types of master integrals appearing in the calculation are obtained analytically for the first time. We get the short-distance coefficient of the leading matrix element to order ${\alpha }_s^2$ by matching the full perturbative QCD calculation results to the corresponding NRQCD ones. The result in this work helps the evaluation of the $B_c$ leptonic decay constant, as well as the Cabibbo-Kobayashi-Maskawa matrix element $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$, to the full next-to-next-to-leading order degree of accuracy.