  We present the next-to-next-to-leading order (NNLO) QCD corrections to the
decay constants for both the pseudoscalar and vector beauty-charmed mesons
$B_{c}$ and $B^*_{c}$ in nonrelativistic QCD effective theory. Explicit NNLO
calculation verified that the $B_c$ decay constant from pseudoscalar current is
identical with the $B_c$ decay constant from axial-vector current. The NNLO
result for the vector decay constant of $B^*_{c}$ meson is novel. Combined with
the latest extraction of nonrelativistic QCD long-distance matrix elements of
$B_c$ meson, we give the branching ratios of leptonic decays of $B_{c}$ and
$B^*_{c}$ mesons. In addition, the novel anomalous dimension for the
flavor-changing heavy quark vector current in nonrelativistic QCD effective
theory are helpful to investigate the threshold behaviours of two different
heavy quarks.
