A fast and precise method to solve the Altarelli-Parisi equations in x space

TL;DR

The paper presents a fast, precise x-space solver for the Altarelli-Parisi evolution equations using a finite-element/ Deboor-inspired discretization with linear (Lagrange) basis functions. By reformulating the AP system as a band-matrix evolution and exploiting a scaling property of convolution kernels, it achieves an explicit finite-series solution with $O(n^2)$ computational cost per step and precomputed integrals. It addresses non-commutativity corrections in NLLA kernels, showing that point-to-point grid evolution largely renders these corrections negligible for practical purposes. Precision studies reveal robust low-$x$ accuracy (≈0.5% with moderate $n$) and effective high-$x$ handling via a two-net $x$-grid, preserving the momentum sum-rule to within $2 imes10^{-4}$ and enabling substantial speedups for global QCD analyses.

Abstract

A numerical method to solve linear integro-differential equations is presented. This method has been used to solve the QCD Altarelli-Parisi evolution equations within the H1 Collaboration at DESY-Hamburg. Mathematical aspects and numerical approximations are described. The precision of the method is discussed.

Figures (4)

• Figure 1: Non commutative corrections, see text.
• Figure 2: a) $R_{q_{ NS}}(x,n)$; b) $R_\Sigma(x,n)$; c) $R_g(x,n)$. See eq. (\ref{['r']}) for definitions. The less accurate curve (far from the line $R=1$) corresponds to $n=100$. The next one to $n=200$ etc...
• Figure 3: a) $R'_{q_{ NS}}(x,n,n')$; b) $R'_\Sigma(x,n,n')$; c) $R'_g(x,n,n')$. See eq. (\ref{['rprime']}) for definitions. The full lines correspond to $n=700$ and the dashed lines correspond to $n=250$. The less accurate curve (far from the line $R=1$) corresponds to $n'=1$. The next one to $n'=4$. The last one (defined only in the case $n=250$) to $n'=8$.
• Figure 4: Momentum sum-rule as function of $Q^2$. The curves described in the text superpose almost exactly.