  We investigate the form factors of $B_c$ meson into $S$-wave charmonium
within the nonrelativistic QCD effective theory and obtain the next-to-leading
order relativistic corrections to the form factors, where both the $B_c$ meson
and the charmonium are treated as the nonrelativistic bound states. Treating
the charm quark as a light quark in the limit \ $m_c/m_b\to 0$, some form
factors are identical at the maximum recoil point, which are consistent with
the predictions in the heavy-quark effective theory and the large-energy
effective theory. Considering that the branching ratios of $B^+_c \to J/\psi
D_s^+$ and $B^+_c \to J/\psi D_s^{*+}$ have been measured by the LHCb and ATLAS
Collaborations recently, we employ the form factors of $B_c$ meson into
$S$-wave charmonium at the next-to-leading order accuracy to these two decay
channels and obtain more precise predictions of their decay rates. Numerical
results indicate that the factorizable diagrams dominate the contribution in
these two channels, while the colour-suppressed and the annihilation diagrams
contribute less than 10 percent. Our results are consistent with the LHCb and
ATLAS data.

High Energy Physics - Phenomenology;High Energy Physics - Experiment