Solving the Altarelli–Parisi equations with truncated moments

Abstract

The technique of truncated moments of parton distributions allows us to study scaling violations without making any assumption on the shape of parton distributions. The numerical implementation of the method is however difficult, since the evolution equations for truncated moments are not diagonal. We present a simple way to improve the efficiency of the numerical solution of the evolution equations for truncated moments. As a result, the number of truncated moments needed to achieve the required precision in the evolution is significantly smaller than in the original formulation of the technique. The method presented here can also be used to obtain the value of parton distributions in terms of truncated moments, and therefore it can be viewed as a technique for the solution of the Altarelli-Parisi equations.

Cut moments approach in the analysis of DIS data

“…This circumstance can be the main reason for large uncertainties at data processing: this effect is aggravated if a singularity of f (x, µ 2 ) in the neighborhood of x = 0 is expected [6]. The idea of "truncated" Mellin moments of the parton densities in QCD analysis was introduced and developed in the late 1990s [7][8][9][10]. The authors obtained the nondiagonal differential evolution equations, in which the nth truncated moment couples to all higher ones.…”

QCD Analysis of Hadron Multiplicities from e+e−-annihilation and Determination of Fragmentation Functions

“…This can now be solved using a numerical technique of your choice, thereby reducing the numerical integrals to solve from three to one and cutting down on computing time significantly [13][14][15].…”

Analysis of DIS structure functions of the nucleon within truncated Mellin moments approach

“…Then, Mellin moments of the parton distributions and structure functions (SFs), which are essential in testing sum rules, are obtained as integrals of the distribution or structure functions over the Bjorken-x variable. An alternative approach, in which one can study directly the evolution of the truncated moments of the parton distributions was proposed in [6], [7], [8], [9]. Later on, we elaborated the exact evolution equations for the truncated moments of the parton densities and structure functions [10], [11], [12].…”

“…The similarity of the evolution equations for the TMM, Eqs. (6)-(8), to the ordinary DGLAP for PDFs, Eqs. (2)-(4), enables one to use the same methods of solving in both the cases.…”