  A hyper-accreting stellar-mass black hole has been long speculated as the
best candidate of central engine of gamma-ray bursts (GRBs). Recent rich
observations of GRBs by space missions such as Swift and Fermi pose new
constraints on GRB central engine models. In this paper, we study the baryon
loading processes of a GRB jet launched from a black hole central engine. We
consider a relativistic jet powered by $\nu \bar\nu$-annihilation or by the
Blandford-Znajek (BZ) mechanism. We consider baryon loading from a
neutrino-driven wind from a neutrino-cooling-dominated accretion flow. For a
magnetically dominated BZ jet, we consider neutron-drifting from the magnetic
wall surrounding the jet and subsequent positron capture and proton-neutron
inelastic collisions. The minumim baryon loads in both types of jet are
calculated. We find that in both cases, a more luminous jet tends to be more
baryon poor. A neutrino-driven "fireball" is typically "dirtier" than a
magnetically dominated jet, while a magnetically dominated jet can be much
cleaner. Both models have the right scaling to interpret the empirical
$\Gamma-L_{\rm iso}$ relation discovered recently. Since some neutrino-driven
jets have too much baryon loading as compared with the data, we suggest that at
least a good fraction of GRBs should have a magnetically dominated central
engine.
