  We present for the first time the complete twist-4 result for the
semi-inclusive deeply inelastic scattering $e^- N \to e^-qX$ with polarized
electron and proton beams at the tree level of pQCD. The calculations have been
carried out using the formulism obtained after collinear expansion where the
multiple gluon scattering are taken into account and gauge links are obtained
automatically in a systematical way. The results show in particular that there
are twist-4 contributions to all the eight twist-2 structure functions for $e^-
N \to e^-hX$ that correspond to the eight twist-2 transverse momentum dependent
parton distribution functions. Such higher twist effects could be very
significant and therefore have important impacts on extracting these
three-dimensional parton distribution functions from the asymmetry data on $e^-
N \to e^-hX$. We suggest also an approximate way for a rough estimation of such
higher twist contributions.
