  In this paper, we study the light scalar and pseudoscalar invisible particles
in the flavor changing neutral current processes of the $B_c$ meson. Effective
operators are introduced to describe the couplings between quarks and light
invisible particles. The Wilson coefficients are extracted from the
experimental results of the $B$ and $D$ mesons, which are used to predict the
upper limits of the branching fractions of the similar decay processes for the
$B_c$ meson. The hadronic transition matrix element is calculated with the
instantaneously approximated Bethe-Salpeter method. The upper limits of the
branching fractions when $m_\chi$ taking different values are presented. It is
found that at some region of $m_\chi$, the channel $B_c\to
D_s^{(\ast)}\chi\chi$ has the largest upper limit which is of the order of
$10^{-6}$, and for $B_c\to D_s^\ast\chi\chi^\dagger$, the largest value of the
upper limits can achieve the order of $10^{-5}$. Other decay modes, such as
$B_c\to D^{(*)}\chi\chi^{(\dagger)}$ and $B_c\to B^{(*)}\chi\chi^{(\dagger)}$,
are also considered.
