Eur. Phys. J. C (2011) 71: 1806
https://doi.org/10.1140/epjc/s10052-011-1806-z

Regular Article - Theoretical Physics

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

¹ Department of Physics and Astronomy, Northwestern University, Evanston, IL, 60208, USA
² Department of Physics, University of Wisconsin, Madison, WI, 53706, USA
³ 415 Pearl Ct., Aspen, CO, 81611, USA

^* e-mail: ldurand@hep.wisc.edu

Received: 15 September 2011
Revised: 7 November 2011
Published online: 19 November 2011

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 . 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)→0 less rapidly than 1/s as s→∞, e.g., as 1/s ^β for 0<β<1. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of β. The new algorithm is exact if the original function G(v) is given by the product of a power v ^β−1 and a polynomial in v. We test the algorithm numerically for very small positive β, β=10⁻⁶ obtaining numerical results that imitate the Dirac delta function δ(v). We also devolve the published MSTW2008LO gluon distribution at virtuality Q ²=5 GeV² down to the lower virtuality Q ²=1.69 GeV². For devolution, β 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.