Analytic solution to leading order coupled DGLAP evolution equations: A new perturbative QCD tool

Abstract

We have analytically solved the LO perturbative QCD singlet DGLAP equations [V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972)][G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977)][Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977)] using Laplace transform techniques. Newly developed, highly accurate, numerical inverse Laplace transform algorithms [M. M. Block, Eur. Phys. J. C 65, 1 (2010)][M. M. Block, Eur. Phys. J. C 68, 683 (2010)] allow us to write fully decoupled solutions for the singlet structure function ${F}_{s}(x,{Q}^{2})$ and $G(x,{Q}^{2})$ as ${F}_{s}(x,{Q}^{2})={\mathcal{F}}_{s}({F}_{s0}({x}_{0}),{G}_{0}({x}_{0}))$ and $G(x,{Q}^{2})=\mathcal{G}({F}_{s0}({x}_{0}),{G}_{0}({x}_{0}))$, where the ${x}_{0}$ are the Bjorken $x$ values at ${Q}_{0}^{2}$. Here ${\mathcal{F}}_{s}$ and $\mathcal{G}$ are known functions---found using LO DGLAP splitting functions---of the initial boundary conditions ${F}_{s0}(x)\ensuremath{\equiv}{F}_{s}(x,{Q}_{0}^{2})$ and ${G}_{0}(x)\ensuremath{\equiv}G(x,{Q}_{0}^{2})$, i.e., the chosen starting functions at the virtuality ${Q}_{0}^{2}$. For both $G(x)$ and ${F}_{s}(x)$, we are able to either devolve or evolve each separately and rapidly, with very high numerical accuracy---a computational fractional precision of $O({10}^{\ensuremath{-}9})$. Armed with this powerful new tool in the perturbative QCD arsenal, we compare our numerical results from the above equations with the published MSTW2008 and CTEQ6L LO gluon and singlet ${F}_{s}$ distributions [A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C 63, 189 (2009)], starting from their initial values at ${Q}_{0}^{2}=1\text{ }\text{ }{\mathrm{GeV}}^{2}$ and $1.69\text{ }\text{ }{\mathrm{GeV}}^{2}$, respectively, using their choice of ${\ensuremath{\alpha}}_{s}({Q}^{2})$. This allows an important independent check on the accuracies of their evolution codes and, therefore, the computational accuracies of their published parton distributions. Our method completely decouples the two LO distributions, at the same time guaranteeing 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 on the evolved gluon and singlet structure functions, as functions of both ${Q}^{2}$ and ${Q}_{0}^{2}$, being equally accurate in devolution (${Q}^{2}<{Q}_{0}^{2}$) as in evolution (${Q}^{2}>{Q}_{0}^{2}$). Further, it can also be used for nonsinglet distributions, thus giving LO analytic solutions for individual quark and gluon distributions at a given $x$ and ${Q}^{2}$, rather than the numerical solutions of the coupled integral-differential equations on a large, but fixed, two-dimensional grid that are currently available.