Longitudinal Structure Function at the Limit $x=Q^2/s$

The longitudinal structure function for nucleons and nuclei is considered at fixed $\sqrt{s}$ and $Q^2$ to the minimum value of $x$ given by $Q^2/s$. This is done using the expansion method and color dipole model in the next-to-leading order approximation. The extracted longitudinal structure functions were consistent with HERA hepdata [ https://www.hepdata.net/record/ins1377206] and the determination of $F_{L}$ at $x=Q^2/s$ [Frank E.Taylor, Phys. Rev. D {\bf111}, 052001 (2025)] at moderate and large $Q^2$ values. The results, consistent with the dipole picture at low $Q^2$ values, show that the longitudinal structure function is small as expected due to the transverse polarization of the exchanged photon and the strong suppression of the dominant gluon component. Nonlinear corrections to the nuclear longitudinal structure function at low values of $x$ and $Q^2$ are also considered. These results may enhance the deep inelastic scattering neutral current data in future colliders at low $x$ and low $Q^2$. The longitudinal structure functions for deuterium at low four-momentum transfer squared, $Q^2<1~\mathrm{GeV}^2$ at the LO and NLO approximations are determined and compared with the JLab E00-002 data.