  We calculate the neutral current jet production semi-inclusive deeply
inelastic scattering process in this paper. Neutral current implies that
interactions can be mediated by the photon, $Z^0$-boson and their interference.
The initial electron is assumed to be polarized and then scattered off by a
target particle with spin-1/2. Calculations are carried out up to twist-3 level
in the quantum chromodynamics parton model by applying the collinear expansion
formalism where multiple gluon scattering is taken into account and gauge links
are obtained automatically. After obtaining the differential cross section, we
introduce the definition of the intrinsic asymmetry. This quantity reveals the
asymmetry in the distribution of the quark intrinsic transverse momentum. We
find that these asymmetries can be expressed in terms of the transverse
momentum dependent parton distribution functions and the electroweak couplings.
As a result, our calculations provide a set of new quantities for analyzing the
parton distribution functions and the electroweak couplings. It is helpful to
understand the hadronic weak interactions and strong interactions in the deeply
inelastic scattering process simultaneously.
