A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

TL;DR

This paper introduces a new numerical algorithm for inverse Laplace transforms applicable to gluon distribution calculations, especially for cases where the original method fails, including negative and non-integer powers.

Contribution

The authors develop a novel, exact numerical algorithm for inverse Laplace transforms that works for all positive and non-integer negative powers, extending previous methods.

Findings

The new algorithm accurately handles cases with small positive eta, mimicking the Dirac delta function.

Successfully devolved the MSTW2008LO gluon distribution to lower virtuality Q^2 values.

Introduced the concept of Hadamard Finite Part integrals for inverse transforms that are distributions.

Abstract

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function $F_2^{\gamma p}(x,Q^2)$. We numerically inverted the function $g(s)$, $s$ being the variable in Laplace space, to $G(v)$, where $v$ is the variable in ordinary space. We have since discovered that the algorithm does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$ as $s\rightarrow\infty$, e.g., as $1/s^\beta$ for $0<\beta<1$. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of $\beta$. The new algorithm is {\em exact} if the original function $G(v)$ is given by the product of a power $v^{\beta-1}$ and a polynomial in $v$. We test the algorithm numerically for very small positive $\beta$, $\beta=10^{-6}$ obtaining numerical results that imitate the Dirac delta function $\delta(v)$. We also devolve the published MSTW2008LO gluon distribution at virtuality $Q^2=5$ GeV$^2$ down to the lower virtuality $Q^2=1.69$ GeV$^2$. For devolution, $ \beta$ is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.