Higher order approximations to the longitudinal structure function $F_{L}$ from the parametrization of $F_{2}$ based on the Laplace transformation

We describe the determination of the longitudinal structure function $F_{L}$ at NLO and NNLO approximations, using Laplace transform techniques, into the parametrization of $F_{2}(x,Q^{2})$ and its derivative with respect to $\ln{Q^{2}}$ at low values of the Bjorken variable $x$. The obtained results are comparable with others by considering the effect of the charm quark mass to the longitudinal structure function, which leads to rescaling variable for $n_{f}=4$. Numerical calculations and comparison with H1 data demonstrate that the suggested method provides reliable $F_{L}(x,Q^{2})$ at small $x$ in a wide range of $Q^{2}$ values and can be applied as well in analyses of ultra-high energy processes with cosmic neutrinos. The obtained longitudinal structure functions with and without the LHeC simulated uncertainties [CERN-ACC-Note-2020-0002, LHeC Collaboration and FCC-he Study Group, P. Agostini et al., J. Phys. G: Nucl. Part. Phys. {\bf48}, 110501(2021).] are compared with the H1 Collaboration data [Eur.Phys.J.C{\bf74}, 2814(2014) and Eur.Phys.J.C{\bf71}, 1579 (2011)] and with the results from CT18 [Phys.Rev.D{\bf103}, 014013(2021)] parametrization model at NLO and NNLO approximations.

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