A determination of the longitudinal structure function $F_{L}$ from the parametrization of $F_{2}$ based on the Laplace transformation

Abstract

I calculate the longitudinal structure function, using Laplace transform techniques, from the parametrization of the structure function $F_{2}(x,Q^{2})$ and its derivative at low values of the Bjorken variable x. I consider the effect of the charm quark mass to the longitudinal structure function, which leads to rescaling variable for $n_{f}=4$. The results are compared with the H1 collaboration data and the CTEQ-Jefferson Lab (CJ) collaboration (Accardi et al. in Phys Rev D 93:114017, 2016) parametrization model. The obtained results with the Bjorken variable of x are found to be comparable with the results Kaptari et al. (Phys Rev D 99:096019, 2019) which is based on the Mellin transform techniques.

Appendix A

The proton structure function parameterized in Ref. [6] provides good fits to the HERA data at low x and large $Q^{2}$ values. The explicit expression for the proton structure function, with respect to the Block–Halzen fit, in a range of the kinematical variables x and $Q^{2}$, $x\le 0.1$ and $0.15\,\mathrm {GeV}^{2}<Q^{2}<3000\,\mathrm {GeV}^{2}$, is defined by the following form

$$\begin{aligned} F^{\gamma p}_{ 2}(x,Q^{2})&= D(Q^{2})(1- x)^{n}\left[ C(Q^{2})+A(Q^{2})\ln \left( \frac{1}{x}\frac{Q^{2}}{Q^{2}+\mu ^{2}}\right) \right. \nonumber \\&\quad +\left. B(Q^{2})\ln ^{2}\left( \frac{1}{x}\frac{Q^{2}}{Q^{2}+\mu ^{2}}\right) \right] , \end{aligned}$$

(11)

Table 1 The effective parameters at low x for $0.15\,\mathrm {GeV}^{2}<Q^{2}<3000\,\mathrm {GeV}^{2}$ provided by the following values

where

$$\begin{aligned} A(Q^{2})= & {} a_{0} + a_{1} {\ln }\left( 1+\frac{Q^{2}}{\mu ^{2}}\right) + a_{2}{\ln }^{2}\left( 1+\frac{Q^{2}}{\mu ^{2}}\right) ,\nonumber \\ B(Q^{2})= & {} b_{0} + b_{1} {\ln }(1+\frac{Q^{2}}{\mu ^{2}}) + b_{2}{\ln }^{2}\left( 1+\frac{Q^{2}}{\mu ^{2}}\right) ,\nonumber \\ C(Q^{2})= & {} c_{0} + c_{1} {\ln }\left( 1+\frac{Q^{2}}{\mu ^{2}}\right) ,\nonumber \\ D(Q^{2})= & {} \frac{Q^{2}(Q^{2}+\lambda M^{2})}{(Q^{2}+M^{2})^2}. \end{aligned}$$

(12)

Here, M is the effective mass and $\mu ^{2}$ is a scale factor. The additional parameters with their statistical errors are given in Table 1.