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

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

Figure 1

The longitudinal structure functions $F_L(x,Q^2)$ [up triangle ($N_f=4$)], down triangle ($N_f=3+\mathrm{Heavy}$) compared by H1 [1] (circles) at the given values of $Q^2$ accompanied by total uncertainties. The determined error bars represent the derivative of $F_2(x,Q^2)$ uncertainties. The curves represent the prediction from the expanding of gluon behavior [3, 4, 5, 6].

Figure 2

The same as Fig. 1 at the given values of $Q^2$ for $N_f=4$ and $N_f=3$.

Figure 3

The longitudinal structure function $F_L$ compared with H1 data averaged over $x$ in the region $6.5\le Q^2\le 800{\text{GeV}}^2$ (solid points). The error bars represent the full errors as obtained by the Monte Carlo procedure described in the Ref. [1]. For each $Q^2$ the average value of $x$ is given above each data point.

Figure 4

The high-order corrections to the gluonic longitudinal structure function compared with H1 data [1].

Figure 5

Nonlinear corrections (NLCs) to the gluonic longitudinal structure function $F_L$ at LO analysis for $N_f=4$ at $R=2{\mathrm{GeV}}^{-1}$ compared with H1 data at $Q^2=6.5$ and 20 ${\mathrm{GeV}}^2$ (solid points).

Figure 6

High-order nonlinear corrections to the gluonic longitudinal structure function $F_L$ at NLO up to NNLO analysis for $N_f=4$ at $R=2{\mathrm{GeV}}^{-1}$ compared with H1 data at $Q^2=6.5$, 20, and 200 ${\mathrm{GeV}}^2$ (solid points).

Figure 7

NNLO nonlinear corrections to the gluonic longitudinal structure function $F_L$ for $N_f=4$ at $R=4{\mathrm{GeV}}^{-1}$ compared with H1 data at $Q^2=6.5$ and 20 ${\mathrm{GeV}}^2$ (solid points).