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 FL(x,Q²), at the leading-order approximation in momentum space, from the structure function F₂(x,Q²) and its derivative with respect to lnQ² in a kinematical region of low values of the Bjorken variable x. Since the x dependence of F₂(x,Q²) and its evolution with Q² are determined much better by the data than FL(x,Q²), this method provides both a direct check on FL(x,Q²) where measured, and a way of extending FL(x,Q²) into regions of x and Q² where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function F₂(x,Q²) [M. M. Block, L. Durand and P. Ha, 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 FL(x,Q²) and that our obtained results are in line with data from the H1 Collaboration and other results for FL(x,Q²) obtained using Mellin transform method.