We describe the determination of the DIS structure functions $F_{ 2}$ and $F_{ L}$ by using the singlet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) and Altarelli-Martinelli equations at small values of $x$. The determination of the longitudinal structure function is presented as a parameterization of $F_{ 2}(x,Q^{ 2})$ and its derivative. Analytical expressions for $\sigma_{ r}(x,Q^{ 2})$ in terms of the effective parameters of the parameterization of $F_{ 2}(x,Q^{ 2})$ and $F_{ L}(x,Q^{ 2})$ are presented. This analysis is enriched by including the higher-twist effects in calculation of the reduced cross sections which is important at low-$x$ and low-$Q^{ 2}$ regions. Numerical calculations and comparison with H1 data demonstrate that the suggested method provides reliable $F_{ L}(x,Q^{ 2})$ and $\sigma_{ r}(x,Q^{ 2})$ at low $x$ in a wide range of the low absolute four-momentum transfers squared ($1.5~\mathrm{ GeV}^{ 2}<Q^{ 2}<120~\mathrm{ GeV}^{ 2}$) at moderate and high inelasticity. Expanding the method to low and ultra low values of $x$ can be considered in the process analysis of new colliders. We compare the obtained longitudinal structure function with respect to the LHeC simulated uncertainties [CERN-ACC-Note-2020-0002, arXiv:2007.14491 [hep-ex] (2020)] with the results from CT18 [Phys.Rev.D{ \bf103}, 014013(2021)] parametrization model.
Categories: hep-ph