2010/10/31 by M. M. Block, Martin M. Block, Loyal Durand +2 · 1 citation
Mathematics · Physics and Astronomy · #DGLAP #Function (biology) #Geometry #Gluon #High-Energy Particle Collisions Research #Inverse #Laplace transform #Mathematical analysis #Mathematical physics #Mathematics #Order (exchange) #Particle physics #Particle physics theoretical and experimental studies #Parton #Perturbative QCD #Physics #Quantum Chromodynamics and Particle Interactions #Quantum chromodynamics #Quantum mechanics #Singlet state #hep-ph
paper · pdf · doi:10.1103/physrevd.83.054009
published as Phys.Rev.D83:054009,2011 · 13 pages, 5 figures, typos corrected, references updated and a footnote added; Accepted for publication in Physical Review D
arxiv created 2011/02/05 · openalex publication_date 2011/03/07 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Fs(x,Q2) and G(x,Q2) as Fs(x,Q2)=Fs(Fs0(x0),G0(x0)) and G(x,Q2)=G(Fs0(x0),G0(x0)), where the x0 are the Bjorken x values at Q02. Here Fs and G are known functions---found using LO DGLAP splitting functions---of the initial boundary conditions Fs0(x)\ensuremath≡Fs(x,Q02) and G0(x)\ensuremath≡G(x,Q02), i.e., the chosen starting functions at the virtuality Q02. For both G(x) and Fs(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 Fs 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 Q02=1 GeV2 and 1.69 GeV2, respectively, using their choice of \ensuremathαs(Q2). 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 Fs 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 Q2 and Q02, being equally accurate in devolution (Q2<Q02) as in evolution (Q2>Q02). 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 Q2, rather than the numerical solutions of the coupled integral-differential equations on a large, but fixed, two-dimensional grid that are currently available.