  With the discovery of the doubly heavy $\Xi_{cc}$ baryon, comprehensive
studies of the properties of the doubly heavy baryons are started. In the
present work, we examine the $\Xi_{bb} \to \Sigma_b \ell^+ \ell^-$ and
$\Xi_{cc} \to \Sigma_c \ell^+ \ell^-$ decays induced by flavor-changing neutral
currents (FCNC) in the framework of the light-cone sum rules. After obtaining
the sum rules for the form factors induced by the tensor current, the branching
ratios of the relevant transitions are estimated. We found that the branching
ratio for the $c \to u$ transition is around five orders smaller than the $b
\to d$ transition. Our findings are also compared with other approaches.
