A Semianalytical Method to Evolve Parton Distributions

TL;DR

This work tackles the fast and accurate solution of DGLAP evolution at NLO in $x$-space by constructing an evolution operator $T(t-t0)$ as a rapidly convergent matrix series based on the splitting functions. The method employs an $x$-grid to convert convolutions into matrix products, yielding a compact, easily reusable operator that evolves any initial polarized parton distribution with minimal CPU time. Numerical tests demonstrate rapid convergence and high accuracy across a wide $x$-range, with significant speed advantages over Laguerre and Mellin-based approaches and no need for moment calculations. The approach provides a practical and scalable tool for analyzing nucleon structure functions in large experimental datasets.

Abstract

We present a new method to solve in a semianalytical way the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at NLO order in the x-space. The method allows to construct an evolution operator expressed in form of a rapidly convergent series of matrices, depending only on the splitting functions. This operator, acting on a generic initial distribution, provides a very accurate solution in a short computer time (only a few hundredth of second). As an example, we apply the method, useful to solve a wide class of systems of integrodifferential equations, to the polarized parton distributions.

Figures (6)

• Figure 1: The initial Singlet distribution ($Q^2=4~GeV^2$, solid line) and the evolved ones for $n=3$ (dashed lines), $n=6$ (dotted lines) and $n=12$ (solid lines) corresponding at $Q^2=200~GeV^2$ and $Q^2=50000~GeV^2$. We use $M=100$.
• Figure 2: The same in Fig. \ref{['f:fig1']} for Gluons.
• Figure 3: The initial Singlet distribution ($Q^2=4~GeV^2$, solid line) and the evolved one at $Q^2=200~GeV^2$ with $M=100$ (solid line), $M=50$ (dotted line) and $M=25$ (dashed line) with $n=12$.
• Figure 4: The same in Fig. \ref{['f:fig3']} for Gluons.
• Figure 5: For $n=12$, $M=100$ and $Q^2 = 200~GeV^2$ both sides of the Eq. (\ref{['e:SG']}) are plotted (solid line and dotted line), in correspondence of the Singlet distribution. Dashed line represents ${\cal E}(x)~\times~10^{3}$ (see text).