  Doubly-heavy baryons, with two heavy and one light quarks, are expected to
exist in QCD and their masses have been predicted in the quark model. However
their existence is not well established so far in experiment. In this work, we
explore the possibility of searching for $\Xi_{bc}$ and $\Xi_{cc}^{+}$ in the
$W$-exchange processes, $\Xi_{bc}^{0}\to pK^{-}$ and $\Xi_{cc}^{+}\to
\Sigma_{c}^{++}(2520)K^{-}$. On the basis of perturbative calculations, we
estimate the branching ratio of the first decay as ${\cal BR}(\Xi_{bc}^0\to p^+
K^-)\approx3.21\times {\cal R}^2_f \times {\cal R}_{\tau}\times 10^{-7}$, where
${\cal R}_f$ (${\cal R}_{\tau}$) are the ratios of the decay constants
(lifetimes) of $\Xi_{bc}^{0}$ and $\Lambda_b^{0}$. The branching ratio of
$\Xi_{cc}^{+}\to \Sigma_{c}^{++}(2520)K^{-}$ is related to that of
$\Lambda_c^+\to \Delta^{++} K^-$, and thereby a conjectured topology analysis
leads to the range for the branching ratio as: ${\cal BR}(\Xi_{cc}^+\to
\Sigma_{c}^{++}(2520) K^-)\in \left[0.36\%,1.80\%\right]$. The decay
$\Xi_{cc}^+\to \Sigma_{c}^{++}(2520) K^-$ would be reconstructed in the $
\Lambda_{c}^{+}K^{-}\pi^{+}$ final state which is easy to access even at a
hadron collider. Based on the two facts that abundant heavy quarks can be
produced at a hadron collider like LHC, and the branching ratios of
$\Xi_{bc}^{0}\to pK^{-}$ and $\Xi_{cc}^{+}\to \Sigma_{c}^{++}(2520)K^{-}$ are
sizable, we urge our experimental colleagues to perform a search at LHCb. This
will presumably lead to the discovery of the $\Xi_{bc}$ and $\Xi_{cc}^{+}$ ,
and precision measurements of the branching ratios in the future are helpful to
investigate their decay mechanism.
