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We study the interface between Regge behavior and DGLAP evolution in a non-perturbative model for the nucleon structure function based on a multipole pomeron exchange. This model provides the input for a subsequent DGLAP evolution that we calculate numerically. The soft input and its evolution give a good fit to the experimental data in the whole available range of x and Q^2.
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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).
We have analytically solved the LO pQCD singlet DGLAP equations using Laplace transform techniques. Newly-developed highly accurate numerical inverse Laplace transform algorithms 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)={cal F}_s(F_{s0}(x), G_0(x)) and G(x,Q^2)={cal G}(F_{s0}(x), G_0(x)). Here {cal F}_s and cal G are known functions of the initial boundary conditions F_{s0}(x) = F_s(x,Q_0^2) and G_{0}(x) = G(x,Q_0^2), i.e., the chosen starting functions at the virtuality Q_0^2. For both G and F_s, we are able to either devolve or evolve each separately and rapidly, with very high numerical accuracy, a computational fractional precision of O(10^{-9}). Armed with this powerful new tool in the pQCD arsenal, we compare our numerical results from the above equations with the published MSTW2008 and CTEQ6L LO gluon and singlet F_s distributions, starting from their initial values at Q_0^2=1 GeV^2 and 1.69 GeV^2, respectively, using their choices of 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 as in evolution. Further, it can also be used for non-singlet 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.
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}) onumber &&mathrm{and} onumber G(x,Q^{2}) &=& mathcal{G}(G_{0}, partial G_{0}). onumber end{eqnarray} We compared our results with the MSTW parameterization and the experimental measurements of $F_{2}^{p}(x,Q^{2})$.
The Regge limit of gauge-theory amplitudes and cross sections is a powerful theory tool for the study of fundamental interactions. It is a vast field of research, encompassing perturbative and non-perturbative dynamics, and ranging from purely theoretical developments to detailed phenomenological applications. It traces its origins to the proposal of Tullio Regge, almost sixty years ago, to study scattering phenomena in the complex angular momentum plane. In this very brief contribution, we look back to the early days of Regge theory, and follow a few of the many strands of its development, reaching to present day applications to scattering amplitudes in non-abelian gauge theories.
DGLAP evolution equations are modified in order to use all the quark families in the full scale range, satisfying kinematical constraints and sumrules, thus having complete continuity for the pdfs and observables. Some consequences of this new approach are shown.
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