  We study ultrafast population dynamics in the topological surface state of
Sb$_2$Te$_3$ in two-dimensional momentum space with time- and angle-resolved
two-photon photoemission. Linear polarized mid-infrared pump pulses are used to
permit a direct optical excitation across the Dirac point. We show that this
resonant excitation is strongly enhanced within the Dirac cone along three of
the six $\bar{\Gamma}$-$\bar{M}$ directions and results in a macroscopic
photocurrent when the plane of incidence is aligned along a
$\bar{\Gamma}$-$\bar{K}$ direction. Our experimental approach makes it possible
to disentangle the decay of transiently excited population and photocurent by
elastic and inelastic electron scattering within the full Dirac cone in
unprecedented detail. This is utilized to show that doping of Sb$_2$Te$_3$ by
vanadium atoms strongly enhances inelastic electron scattering to lower
energies, but only scarcely affects elastic scattering around the Dirac cone.
