A new numerical method for obtaining gluon distribution functions G(x,Q²)=xg(x,Q²), from the proton structure function F₂γᵖ(x,Q²)

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An exact expression for the leading-order (LO) gluon distribution function G(x,Q²)=xg(x,Q²) from the DGLAP evolution equation for the proton structure function F₂γᵖ(x,Q²) for deep inelastic γ* p scattering has recently been obtained [M. M. Block, L. Durand and D. W. McKay, Phys. Rev. D 79, 014031, (2009)] for massless quarks, using Laplace transformation techniques. Here, we develop a fast and accurate numerical inverse Laplace transformation algorithm, required to invert the Laplace transforms needed to evaluate G(x,Q²), and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire x and Q² spectrum. Since no analytic Laplace inversion is possible for next-to-leading order (NLO) and higher orders, this numerical algorithm will enable one to obtain accurate NLO (and NNLO) gluon distributions, using only experimental measurements of F₂γᵖ(x,Q²).