Higher order approximations to the longitudinal structure function $F_L$ from the parametrization of $F_2$ based on the Laplace transformation

Abstract

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.

Comments: Will be appear in Phys.Rev.D