The path to $$\hbox {N}^3\hbox {LO}$$ N 3 LO parton distributions

Abstract We extend the existing leading (LO), next-to-leading (NLO), and next-to-next-to-leading order (NNLO) NNPDF4.0 sets of parton distribution functions (PDFs) to approximate next-to-next-to-next-to-leading order ( $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO ). We construct an approximation to the $$\hbox {N}^3\hbox {LO}$$ N 3 LO splitting functions that includes all available partial information from both fixed-order computations and from small and large x resummation, and estimate the uncertainty on this approximation by varying the set of basis functions used to construct the approximation. We include known $$\hbox {N}^3\hbox {LO}$$ N 3 LO corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to $$\mathcal {O}\left( \alpha _s^3\right) $$ O α s 3 accuracy. We determine a set of $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO PDFs by accounting both for the uncertainty on splitting functions due to the incomplete knowledge of $$\hbox {N}^3\hbox {LO}$$ N 3 LO terms, and to the uncertainty related to missing higher corrections (MHOU), estimated by scale variation, through a theory covariance matrix formalism. We assess the perturbative stability of the resulting PDFs, we study the impact of MHOUs on them, and we compare our results to the $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO PDFs from the MSHT group. We examine the phenomenological impact of $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO corrections on parton luminosities at the LHC, and give a first assessment of the impact of $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO PDFs on the Higgs and Drell–Yan total production cross-sections. We find that the $$\hbox {aN}^3\hbox {LO}$$ aN 3 LO NNPDF4.0 PDFs are consistent within uncertainties with their NNLO counterparts, that they improve the description of the global dataset and the perturbative convergence of Higgs and Drell–Yan cross-sections, and that MHOUs on PDFs decrease substantially with the increase of perturbative order.