The path to $$\hbox {N}^3\hbox {LO}$$ parton distributions
Richard D. Ball, Andrea Barontini, Alessandro Candido, Stefano Carrazza, Juan Cruz–Martinez, Luigi Del Debbio, Stefano Forte, Tommaso Giani, Felix Hekhorn, Zahari Kassabov, Niccolò Laurenti, Giacomo Magni, Emanuele R. Nocera, Tanjona R. Rabemananjara, Juan Rojo, Christopher Schwan, Roy Stegeman, Maria Ubiali
Abstract
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 ( $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> ). We construct an approximation to the $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>N</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> 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 $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>N</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to $$\mathcal {O}\left( \alpha _s^3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mfenced><mml:msubsup><mml:mi>α</mml:mi><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:msubsup></mml:mfenced></mml:mrow></mml:math> accuracy. We determine a set of $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> PDFs by accounting both for the uncertainty on splitting functions due to the incomplete knowledge of $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>N</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> 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 $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> PDFs from the MSHT group. We examine the phenomenological impact of $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> corrections on parton luminosities at the LHC, and give a first assessment of the impact of $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> PDFs on the Higgs and Drell–Yan total production cross-sections. We find that the $$\hbox {aN}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mtext>aN</mml:mtext><mml:mn>3</mml:mn></mml:msup><mml:mtext>LO</mml:mtext></mml:mrow></mml:math> 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.