Twist-4 contributions to semi-inclusive deeply inelastic scatterings with polarized beam and target

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.