Abstract
Using Laplace transform techniques, I calculate the longitudinal structure function ${F}_{L}(x,{Q}^{2})$ from the scaling violations of the proton structure function ${F}_{2}(x,{Q}^{2})$ and make a critical study of this relationship between the structure functions at leading order (LO) up to next-to-next-to leading order (NNLO) analysis at small $x$. Furthermore, I consider heavy quark contributions to the relation between the structure functions, which leads to compact formula for ${N}_{f}=3+\mathrm{Heavy}$. The nonlinear corrections to the longitudinal structure function at LO up to NNLO analysis are shown in the ${N}_{f}=4$ (light quark flavor) based on the nonlinear corrections at $R=2$ and $R=4\phantom{\rule{4pt}{0ex}}{\text{GeV}}^{\ensuremath{-}1}$. The results are compared with experimental data of the longitudinal proton structure function ${F}_{L}$ in the range of $6.5\ensuremath{\le}{Q}^{2}\ensuremath{\le}800\phantom{\rule{4.pt}{0ex}}{\text{GeV}}^{2}$.
Highlights
The inclusive deep inelastic scattering (DIS) measurements are of importance to understanding the gluonic substructure of proton at low values of the Bjorken variable x
The longitudinal structure function is determined by measur√ements of differential cross sections at different values of s at the Hadron-Ele√ctron Ring Accelerator (HERA) at DESY, where data on s for electron beam energies of Ee 27.5 GeV and for proton beam energies of Ep = 920, 820, 575, and 460 GeV are collected [1,2]
The longitudinal structure functions determined for four massless quarks at m2c < μ2 and to account for fixed the Nf = 3 flavor number scheme as the heavy-flavor contributions to FL are taken as given by fixed order perturbation theory
Summary
The inclusive deep inelastic scattering (DIS) measurements are of importance to understanding the gluonic substructure of proton at low values of the Bjorken variable x. The latter case corresponds to the low values of the Bjorken variable x, and the longitudinal structure function is related to the gluon density of the proton. [3] suggested an approximate relation between the gluon density at the point 2.5x and the longitudinal structure function FL at the point x in the following form: FL(x,Q2). Two different methods were suggested [5,6], the derivatives of the structure functions were based in the expansion of the gluon distribution around the arbitrary point z = α. In Appendix A, I present the results for the splitting functions and coefficients in the inverse Laplace transform method at some values of Q2. Appendix E deals with a technical detail including the inverse Laplace transform of the nonlinear kernels at LO and high-order corrections presented in Appendix F
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