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

摘要

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~(γp)(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)→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 GeV ~2 down to the lower virtuality Q ~2 =1.69 GeV ~2. 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.