Analytic parametrisations of the scale dependence of proton parton distributions

Abstract

Parton distribution functions (PDFs) are non-perturbative inputs whose scale dependence is governed by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. Their numerical evaluation commonly relies on interpolation of tabulated grids. We discuss an analytic representation of collinear proton PDFs in $(x,Q^2)$, obtained by promoting the parameters of a standard $x$-space parametrisation to functions of the scale. The coefficients are reconstructed from the HERAPDF2.0 NLO set using machine-assisted fitting techniques, with sampling in $10^{-4}\le x\le 0.65$ and scales up to 1 TeV. The resulting ML-PDFs reproduce integrated distributions at the sub-percent level for most flavours, with deviations reaching about 1.5% for the strange quark at the highest scales. They satisfy valence and momentum sum rules at the percent level and reproduce representative $pp\to \pi$ and $pp\to \gamma \pi$ cross sections within approximately 2% and 5%, respectively, in the kinematic regions studied. Compared with grid interpolation through LHAPDF, direct evaluation of the analytic expressions reduces the total Monte Carlo run time by about 11% and 30% in these two applications.

arXiv comment6 pages, 3 figures, 1 table, contribution to the proceedings of the 43rd International Conference on High Energy Physics (ICHEP 2026), 30 July - 5 August 2026, Natal, Brazil