  In this work, we evaluate the lifetimes of the doubly charmed baryons
$\Xi_{cc}^{+}$, $\Xi_{cc}^{++}$ and $\Omega_{cc}^{+}$. We carefully calculate
the non-spectator contributions at the quark level where the Cabibbo-suppressed
diagrams are also included. The hadronic matrix elements are evaluated in the
simple non-relativistic harmonic oscillator model. Our numerical results are
generally consistent with that obtained by other authors who used the diquark
model. However, all the theoretical predictions on the lifetimes are one order
larger than the upper limit set by the recent SELEX measurement. This
discrepancy would be clarified by the future experiment, if more accurate
experiment still confirms the value of the SELEX collaboration, there must be
some unknown mechanism to be explored.
