Decoupling of the DGLAP evolution equations by Laplace method

TL;DR

This paper introduces a Laplace transform method to decouple and solve the DGLAP evolution equations for gluon and singlet distributions, providing analytical solutions and comparing them with experimental data.

Contribution

It derives second-order differential equations for gluon and singlet distributions and decouples their solutions based on initial conditions, offering a new analytical approach.

Findings

Solutions agree with MSTW parameterization

Results match experimental measurements of $F_2^p(x,Q^2)$

Decoupling simplifies the analysis of parton distributions

Abstract

In this paper, we derive two second- order of differential equation for the gluon and singlet distribution functions by using the Laplace transform method. We decoupled the solutions of the singlet and gluon distributions into the initial conditions (function and derivative of the function) at the virtuality $Q_{0}^{2}$ separately as these solutions are defined by: \begin{eqnarray} F_{2}^{s}(x,Q^{2}) &=& \mathcal{F}(F_{s0}, \partial F_{s0})\nonumber &&\mathrm{and} \nonumber G(x,Q^{2}) &=& \mathcal{G}(G_{0}, \partial G_{0}).\nonumber \end{eqnarray} We compared our results with the MSTW parameterization and the experimental measurements of $F_{2}^{p}(x,Q^{2})$.