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We present for the first time the complete twist-4 result for thesemi-inclusive deeply inelastic scattering $e^- N \to e^-qX$ with polarizedelectron and proton beams at the tree level of pQCD. The calculations have beencarried out using the formulism obtained after collinear expansion where themultiple gluon scattering are taken into account and gauge links are obtainedautomatically in a systematical way. The results show in particular that thereare 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 dependentparton distribution functions. Such higher twist effects could be verysignificant and therefore have important impacts on extracting thesethree-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 suchhigher twist contributions.

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