  We explore the effects induced by a finite width in processes of pair
production of a heavy top-quark partner and its subsequent decay into a bosonic
Dark Matter (DM) candidate -- either scalar or vector -- and a SM up-type quark
at the Large Hadron Collider (LHC). We discuss the configurations of masses,
widths and couplings where this phenomenology can be important in a simple
model with just one such objects. Finally, we emphasise the correct definition
of signal and background to be adopted as well as stress the importance of new
dedicated experimental searches.
