We have analytically solved the LO pQCD singlet DGLAP equations using Laplacetransform techniques. Newly-developed highly accurate numerical inverse Laplacetransform algorithms allow us to write fully decoupled solutions for thesinglet structure function F_s(x,Q^2)and G(x,Q^2) as F_s(x,Q^2)={\calF}_s(F_{s0}(x), G_0(x)) and G(x,Q^2)={\cal G}(F_{s0}(x), G_0(x)). Here {\calF}_s and \cal G are known functions of the initial boundary conditionsF_{s0}(x) = F_s(x,Q_0^2) and G_{0}(x) = G(x,Q_0^2), i.e., the chosen startingfunctions at the virtuality Q_0^2. For both G and F_s, we are able to eitherdevolve or evolve each separately and rapidly, with very high numericalaccuracy, a computational fractional precision of O(10^{-9}). Armed with thispowerful new tool in the pQCD arsenal, we compare our numerical results fromthe above equations with the published MSTW2008 and CTEQ6L LO gluon and singletF_s distributions, starting from their initial values at Q_0^2=1 GeV^2 and 1.69GeV^2, respectively, using their choices of \alpha_s(Q^2). This allows animportant independent check on the accuracies of their evolution codes andtherefore the computational accuracies of their published parton distributions.Our method completely decouples the two LO distributions, at the same timeguaranteeing that both G and F_s satisfy the singlet coupled DGLAP equations.It also allows one to easily obtain the effects of the starting functions onthe evolved gluon and singlet structure functions, as functions of both Q^2 andQ_0^2, being equally accurate in devolution as in evolution. Further, it canalso be used for non-singlet distributions, thus giving LO analytic solutionsfor individual quark and gluon distributions at a given x and Q^2, rather thanthe numerical solutions of the coupled integral-differential equations on alarge, but fixed, two-dimensional grid that are currently available.
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