  We explore in the mean-field approximation the robustness with respect to
dissipation of self-organized criticality in sandpile models. To this end, we
generalize a recently introduced self-organized branching process, and show
that the model self-organizes not into a critical state but rather into a
subcritical state: when dissipation is present, the dynamical fixed point does
not coincide with the critical point. Thus the level of dissipation acts as a
relevant parameter in the renormalization-group sense. We study the model
numerically and compute analytically the critical exponents for the avalanche
size and lifetime distributions and the scaling exponents for the corresponding
cutoffs.
