Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function $F_L(x,Q^2)$, at the leading-order approximation in momentum space, from the structure function $F_2(x,Q^2)$ and its derivative with respect to $\ln Q^2$ in a kinematical region of low values of the Bjorken variable $x$. Since the $x$ dependence of $F_2(x,Q^2)$ and its evolution with $Q^2$ are determined much better by the data than $F_L(x,Q^2)$, this method provides both a direct check on $F_L(x,Q^2)$ where measured, and a way of extending $F_L(x,Q^2)$ into regions of $x$ and $Q^2$ where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function $F_2(x,Q^2)$ [M. M. Block et al., Phys. Rev. D 89, 094027 (2014).]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function $F_L(x,Q^2)$ and that our obtained results are in line with data from the H1 Collaboration and other results for $F_L(x,Q^2)$ obtained using Mellin transform method.