  In this work, we investigate the semi-leptonic weak decays of spin-1/2
doubly-heavy baryons $\Xi_{QQ^{\prime}}$ into spin-3/2 singly-heavy baryons
$\Sigma_{Q^{\prime}}^*$ within light-cone sum rules. Using the parallel
components of the light-cone distribution amplitudes of
$\Sigma_{Q^{\prime}}^*$, the transition form factors for these decays are
calculated both analytically and numerically. The numerical results for these
semi-leptonic weak decays widths and branching ratios are also predicted, which
are compared with the same predictions by other theoretical approaches in the
literatures. These phenomenology predictions can be tested by the experiments
in the future.
