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Studies of the Deep Inelastic Scattering (DIS) have provided fundamentalinformation of the nucleon structure for decades. The electron-ion collider(EIC) will be the first collider capable of DIS study with both polarizedlepton and polarized hadron beams, providing the possibility of accessing newelectroweak structure functions of the nucleon. In this work, we completed theDIS cross section derivations for both longitudinally and transverselypolarized leptons and hadrons, with no approximations made, and with all threecontributions $\gamma \gamma, \gamma Z, ZZ$ included. These results werederived using primarily tensor algebra and Feynman calculus, starting frompreviously established leptonic and hadronic tensors and carry out theircontraction. Our results are presented in terms of both spin-averaged andspin-dependent cross sections, allowing direct comparison with experimentallymeasured cross sections and their asymmetries. We include also in ourdiscussion comparisons of different conventions that exist in the literature.

Complete Formalism of Cross Sections and Asymmetries for Longitudinally and Transversely Polarized Leptons and Hadrons in Deep Inelastic Scattering