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

Abstract

An exact expression for the leading-order (LO) gluon distribution function G(x,Q 2)=xg(x,Q 2) from the DGLAP evolution equation for the proton structure function $F_{2}^{\gamma p}(x,Q^{2})$ for deep inelastic γ * p scattering has recently been obtained (Block et al., 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,Q 2), and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire x and Q 2 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 $F_{2}^{\gamma p}(x,Q^{2})$ .