  We discuss the discovery potential of a dark force carrier ($Z'$) of very
light mass, $m_{Z'} \lesssim {\cal O}(1-10)$ GeV, at hadron colliders via rare
top quark decays, especially when it decays invisibly in typical search
schemes. We emphasize that the top sector is promising for the discovery of new
particles because top quark pairs are copiously produced at the Large Hadron
Collider. The signal process is initiated by a rare top decay into a bottom
quark and a charged Higgs boson ($H^\pm$) decaying subsequently into a $W$ and
one or multiple $Z'$s. The light $Z'$ can be invisible in collider searches in
various scenarios, and it would be hard to distinguish the relevant collider
signature from the regular $t\bar{t}$ process in the Standard Model. We suggest
a search strategy using the recently proposed on-shell constrained $M_2$
variables. Our signal process is featured by an $\textit{asymmetric}$ event
topology, while the $t\bar{t}$ is $\textit{symmetric}$. The essence behind the
strategy is to evoke some contradiction in the relevant observables by applying
the kinematic variables designed under the assumption of the $t\bar{t}$ event
topology. To see the viability of the proposed technique, we perform Monte
Carlo simulations including realistic effects such as cuts, backgrounds,
detector resolution, and so on at the LHC of $\sqrt{s}=14$ TeV.
