Longitudinal structure function from logarithmic slopes of $F_2$ at low $x$

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 $R=4Ge{V}^{-1}$. The results are compared to experimental data of the longitudinal proton structure function $F_L$ in the range of $6.5\le Q^2\le 800Ge{V}^2$.