A new numerical method for obtaining gluon distribution functions $G(x,Q^2)=xg(x,Q^2)$, from the proton structure function $F_2^{\gamma p}(x,Q^2)$

TL;DR

This paper introduces a fast, precise numerical algorithm for inverting Laplace transforms to extract gluon distribution functions from proton structure data, enabling accurate NLO and NNLO calculations.

Contribution

It develops a highly accurate numerical inverse Laplace transformation method to compute gluon distributions from structure functions, extending applicability beyond leading order.

Findings

Achieves less than 0.1% error in inversion

Enables accurate NLO and NNLO gluon distribution calculations

Provides a practical tool for analyzing deep inelastic scattering data

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 $\gamma^* p$ scattering has recently been obtained [M. M. Block, L. Durand and D. W. McKay, Phys. Rev. D{\bf 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 $F_2^{\gamma p}(x,Q^2)$.