Analytic parametrisations of the scale dependence of proton parton distributions

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,Q²), 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≤ x≤ 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→π and pp→γπ 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.