An evaluation of the proton structure functions F 2 and F L at small x

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 ($1.5{\text{GeV}}^2<Q^2<120{\text{GeV}}^2$) 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 (Agostini et al. (2020) [4]) with the results from CT18 (Hou et al. (2021) [54]) parametrization model.