An Evaluation of The Proton Structure Functions $F_{2}$ and $F_{L}$ at Small $x$

Abstract

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.