Addendum to: "A new numerical method for obtaining gluon distribution functions $G(x,Q^2)=xg(x,Q^2)$, from the proton structure function $F_2^{\gamma p}(x,Q^2)$."

TL;DR

This paper introduces a new numerical algorithm for inverting Laplace transforms that decay as 1/s^β with 0<β<1, addressing limitations of previous methods in calculating gluon distributions from proton structure functions.

Contribution

The authors develop and test a novel numerical inversion algorithm for Laplace transforms that decay slower than 1/s, improving the computation of gluon distributions from structure functions.

Findings

The new algorithm successfully inverts Laplace transforms with decay rates of 1/s^β, 0<β<1.

It enhances the accuracy and applicability of gluon distribution calculations.

The method is validated using the Laplace transform of 1/√v.

Abstract

In a recent Letter entitled "A new numerical method for obtaining gluon distribution functions $G(x,Q^2)=xg(x,Q^2)$, from the proton structure function $F_2^{\gamma p}(x,Q^2)$" [arXiv:0907.4790], we derived an accurate and fast algorithm for numerically inverting Laplace transforms, which we used in obtaining gluon distributions from the proton structure function $F_2^{\gamma p}(x,Q^2)$. We inverted the function $g(s)$, where $s$ is the variable in Laplace space, to $G(v)$, where $v$ is the variable in ordinary space. Since publication, we have discovered that the algorithm does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$, as $s\rightarrow\infty$. Although we require that $g(s)\rightarrow 0$ as $s\rightarrow\infty$, it can approach 0 as ${1\over s^\beta}$, with $0<\beta<1$, and still be a proper Laplace transform. In this note, we derive a new numerical algorithm for just such cases, and test it for $g(s)={\sqrt \pi\over \sqrt s} $, the Laplace transform of ${1\over \sqrt v}$.