  The heaviness of the top quark makes its 2-body decay mode $t \to bW^+$ to be
dominant, at such level that hardly any other decay mode reaches a detectable
branching ratio (BR) within the SM. Here we study the decay $t \to
bW^+\ell^-\ell^+$ ($\ell=e, \mu, \tau$), which diverges for massless leptons,
and it can reach a BR $\sim \mathcal{O}(10^{-5}\sim 10^{-6})$ for reasonable
values of the low energy cut in the lepton-pair invariant mass. This rate
surpasses almost any other rare decays such as $t \to cX$ ($X=\gamma,Z,g,H,
W^+W^-$), and thus offers the possibility of being detectable. Furthermore, the
estimate of this channel is relevant because it can mimic the signal arising
from the lepton number violating decay $t\to bW^-\ell^+\ell^+$, when the $W$
boson decays into lepton channels.
