Higher order approximations to the longitudinal structure function FL from the parametrization of F2 based on the Laplace transformation

G R Boroun,B Rezaei

doi:10.1103/physrevd.105.034002

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.

Highlights

In recent years, many attempts have been made to better understand the longitudinal structure function experimentally and theoretically [1–8]
At extremely small x, the longitudinal structure function becomes predominant, and its behavior will be checked in high energy processes such as the Large Hadron electron Collider (LHeC) and the Future Circular Collider electron
The longitudinal structure function measurement will cover an x range from 2 × 10−6 to above x 1⁄4 0.01 which the LHeC promises to provide, as it extends the kinematic range in electron-proton scattering by nearly four orders of magnitude of ep collisions at HERA [10]

Summary

INTRODUCTION

Many attempts have been made to better understand the longitudinal structure function experimentally and theoretically [1–8]. At extremely small x, the longitudinal structure function becomes predominant, and its behavior will be checked in high energy processes such as the Large Hadron electron Collider (LHeC) and the Future Circular Collider electron-. The longitudinal structure function measurement will cover an x range from 2 × 10−6 to above x 1⁄4 0.01 which the LHeC promises to provide, as it extends the kinematic range in electron-proton (ep) scattering by nearly four orders of magnitude of ep collisions at HERA [10]. The interest in a measurement of the longitudinal structure function, especially at small x, is related to the uncertainty in the determination of the gluon distribution. The longitudinal structure function is directly related to the singlet and gluon distributions in the proton, and its behavior has been predicted by the Altarelli and Martinelli [11] equation. The authors in Ref. [11] derived an elegant formula for the longitudinal structure function FLðx; Q2Þ, an effect of order αsðQ2Þ, as a convolution integral over F2ðx; Q2Þ and the gluon density gðx; Q2Þ by the following form: FLðx; Q2Þ 1⁄4 CL;nsþsðasðQ2Þ; xÞ ⊗ F2ðx; Q2Þ þ he2iCL;gðasðQ2Þ; xÞ ⊗ Gðx; Q2Þ; ð1Þ

Published by the American Physical Society

METHOD

Now let us to continue with the function

RESULTS AND DISCUSSION

The coefficient functions read as