  In this work, we study the weak decay between two heavy baryons $
\mathcal{B}_i(\frac{1}{2}^+)\to \mathcal{B}_f(\frac{3}{2}^+)$ in the
light-front quark model where three-quark picture is employed for baryon. We
derive general form of transition amplitude of $
\mathcal{B}_i(\frac{1}{2}^+)\to \mathcal{B}_f(\frac{3}{2}^+)$, and analyze two
specific cases of transitions: the weak decays of single heavy baryon
$\Sigma_{b} \to \Sigma_{c}^*$ and the decays of double-charmed baryon
$\Xi_{cc}\to \Sigma_{c}^*(\Xi_{c}^*)$. We compute the hadronic form factors for
the transitions and apply them to study the decay widths of the semi-leptonic
$\mathcal B_i(\frac{1}{2}^+)\to\mathcal B_f(\frac{3}{2}^+) l\bar{\nu}_l$ and
non-leptonic $\mathcal B_i(\frac{1}{2}^+)\to\mathcal B_f(\frac{3}{2}^+)M$.
Previously we studied the transition $\Sigma_{b} \to \Sigma_{c}^*$ with the
quark-diquark picture of baryon in the light-front quark model. Here we revisit
this transition with three-quark picture of baryon. At the quark level, the
transition $\Sigma_{b} \to \Sigma_{c}^*$ is induced by the $b\rightarrow c$
transition.The subsystem of the two unchanged light quarks which possesses
definite and same spin in initial and final state can be viewed as a spectator,
so the spectator approximation can be applied directly. For the weak decay of
doubly charmed baryon $\Xi_{cc}$, a $c$ quark decays to a light quark $q_1$, so
both the initial state $cc$ and final state $q_1q_2$ ($q_1$ and the original
$q_2$ in initial state may be the same flavor quarks) which possess definite
spin are no longer spectators. A rearrangement of quarks for initial and final
states is adopted to isolate the unchanged subsystem $cq_2$ which can be viewed
as the spectator approximately. Future measurements on these channels will
constrain the nonperturbative parameter in the wavefunctions and test the model
predictions.
