Parton model contributions as next-to-eikonal corrections to the dipole factorization of DIS and SIDIS at low $x_{Bj}$

We compute the next-to-eikonal (NEik) power corrections to inclusive deep inelastic scattering (DIS) and semi-inclusive deep inelastic scattering (SIDIS) at low $x$ beyond dipole factorization, which represent the eikonal result. The analysis is restricted to contributions arising from t-channel quark exchanges, thereby probing the quark background field of the target. For a transversely polarized virtual photon, the NEik corrections to inclusive DIS are expressed in terms of quark and antiquark collinear parton distribution functions (PDFs), while the corresponding corrections to SIDIS are formulated in terms of the unpolarized quark transverse-momentum-dependent distribution (TMD). In contrast, for a longitudinally polarized photon, the NEik corrections to both inclusive DIS and SIDIS vanish at lowest order in ${\alpha }_s$.