Decoupling the NLO coupled DGLAP evolution equations: an analytic solution to pQCD

Abstract

Using repeated Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we transform the coupled, integral-differential NLO singlet DGLAP equations first into coupled differential equations, then into coupled algebraic equations, which we can solve iteratively. After Laplace inverting the algebraic solution analytically, we numerically invert the solutions of the decoupled differential equations. Finally, we arrive at the decoupled NLO evolved solutions F_s(x,Q^2)=calF_s(F_{s0}(x),G_0(x)) and G(x,Q^2)=calG(F_{s0}(x),G_0(x)), where calF_s and calG are known functions - determined using the DGLAP splitting functions up to NLO in the strong coupling constant alpha_s(Q^2). The functions F_{s0}(x)=F_s(x,Q_0^2) and G_0(x)=G(x,Q_0^2) are the starting functions for the evolution at Q_0^2. This approach furnishes us with a new tool for readily obtaining, independently, the effects of the starting functions on either the evolved gluon or singlet structure functions, as a function of both Q^2 and Q_0^2. It is not necessary to evolve coupled integral-differential equations numerically on a two-dimensional grid, as is currently done. The same approach can be used for NLO non-singlet distributions where it is simpler, only requiring one Laplace transform. We make successful NLO numerical comparisons to two non-singlet distributions, using NLO quark distributions published by the MSTW collaboration, over a large range of x and Q^2. Our method is readily generalized to higher orders in the strong coupling constant alpha_s(Q^2).