The Mellin Transform Technique for the Extraction of the Gluon Density

TL;DR

The paper tackles the challenge of extracting the proton gluon density from jet production in deep inelastic scattering at NLO, where cross sections are non-factorizing in the parton momenta. It introduces a Mellin-transform framework that converts convolutions into products of moments, even when the cross sections depend on more than the simple ratio x_B/ξ, by constructing generalized moments and using an inverse Mellin transform. A practical inversion scheme with contour choices and a controlled z_max is developed, enabling fast and stable numerical evaluations. A detailed HERA-based study demonstrates that the method achieves ~1% accuracy compared to direct calculations, paving the way for efficient gluon-density fits in NLO QCD and applicability to more complex observables.

Abstract

A new method is presented to determine the gluon density in the proton from jet production in deeply inelastic scattering. By using the technique of Mellin transforms not only for the solution of the scale evolution equation of the parton densities but also for the evaluation of scattering cross sections, the gluon density can be extracted in next-to-leading order QCD. The method described in this paper is, however, more general, and can be used in situations where a repeated fast numerical evaluation of scattering cross sections for varying parton distribution functions is required.

Figures (2)

• Figure 1: Generic Feynman diagrams for the leading-order processes of QCD Compton scattering (a) and photon--gluon fusion (b), and an example for a diagram corresponding to a next-to-leading order real correction (c).
• Figure 2: Integration contours of the Mellin inversion in eq. (\ref{['inv']}), leading to the inversion formulae of eqs. (\ref{['inv1']}) and (\ref{['inv2']}) for the routes ${\cal C}_{0}$ and ${\cal C}_{1}$, respectively. The crosses schematically denote the singularities of $F_{n}$.