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

Studies of the Deep Inelastic Scattering (DIS) have provided fundamental information of the nucleon structure for decades. The electron-ion collider (EIC) will be the first collider capable of DIS study with both polarized lepton and polarized hadron beams, providing the possibility of accessing new electroweak structure functions of the nucleon. In this work, we completed the DIS cross section derivations for both longitudinally and transversely polarized leptons and hadrons, with no approximations made, and with all three contributions $\gamma \gamma, \gamma Z, ZZ$ included. These results were derived using primarily tensor algebra and Feynman calculus, starting from previously established leptonic and hadronic tensors and carry out their contraction. Our results are presented in terms of both spin-averaged and spin-dependent cross sections, allowing direct comparison with experimentally measured cross sections and their asymmetries. We include also in our discussion comparisons of different conventions that exist in the literature.