We present a determination of the strong coupling $$\alpha _s(m_Z)$$ from a global dataset including both fixed-target and collider data from deep-inelastic scattering and a variety of hadronic processes, with a simultaneous determination of parton distribution functions (PDFs) based on the NNPDF4.0 methodology. This determination is performed at NNLO and approximate $${\textrm{N}}^3$$ LO ( $${\textrm{aN}}^3$$ LO) perturbative QCD accuracy, including QED corrections and a photon PDF up to NLO accuracy. We extract $$\alpha _s$$ using two independent methodologies, both of which take into account the cross-correlation between $$\alpha _s$$ and the PDFs. The two methodologies are validated by closure tests that allow us to detect and remove or correct for several sources of bias, and lead to mutually consistent results. We account for all correlated experimental uncertainties, as well as correlated theoretical uncertainties related to missing higher order perturbative corrections (MHOUs). We study the perturbative convergence of our results and the impact of QED corrections. We assess individual sources of uncertainty, specifically MHOUs and the value of the top quark mass. We provide a detailed appraisal of methodological choices, including the choice of input dataset, the form of solution of evolution equation, the treatment of the experimental covariance matrix, and the details of Monte Carlo data generation. We find $$\alpha _s(m_Z)=0.1194^{+0.0007}_{-0.0014}$$ at $${\textrm{aN}}^3$$ $${\textrm{LO}}_{\textrm{QCD}}\otimes {\textrm{NLO}}_{\textrm{QED}}$$ accuracy, consistent with the latest PDG average and with recent lattice results.

United Kingdom

Richard Ball

Italy

Andrea Barontini

Switzerland

Juan Cruz-Martinez

Stefano Forte

Finland

Felix Hekhorn

Emanuele Nocera

Netherlands

Juan Rojo

Roy Stegeman

A determination of $$\alpha _s(m_Z)$$ at $${{\textrm{aN}}}^3{{\textrm{LO}}}_{{\textrm{QCD}}}\otimes {{\textrm{NLO}}}_{{\textrm{QED}}}$$ accuracy from a global PDF analysis

We present NNPDF4.0MC, a variant of the NNPDF4.0 set of parton distributions (PDFs) at LO, NLO and NNLO, with and without inclusion of the photon PDF, suitable for use with Monte Carlo (MC) event generators, which require PDFs to satisfy additional constraints in comparison to standard PDF sets. These requirements include PDF positivity down to a low scale Q ~ 1 GeV, smooth extrapolation in the very small and large x regions, and numerically stable results even in extreme regions of phase space for all PDFs. We compare the NNPDF4.0MC PDFs to their baseline NNPDF4.0 counterparts, and to the NNPDF2.3LO set entering the Monash tune of the Pythia8 event generator. We briefly assess the phenomenological impact of these PDFs on the cross-sections for hard and soft QCD processes at the LHC.

CERN

Niccolò Laurenti

Tanjona Rabemananjara

LO, NLO, and NNLO parton distributions for LHC event generators

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 ( $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>  ). We construct an approximation to the  $$\hbox {N}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>N</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </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  $$\hbox {N}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>N</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to  $$\mathcal {O}\left( \alpha _s^3\right) $$  <math> <mrow> <mi>O</mi> <mfenced> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </mfenced> </mrow> </math>   accuracy. We determine a set of  $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   PDFs by accounting both for the uncertainty on splitting functions due to the incomplete knowledge of  $$\hbox {N}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>N</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </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  $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   PDFs from the MSHT group. We examine the phenomenological impact of  $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   corrections on parton luminosities at the LHC, and give a first assessment of the impact of  $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   PDFs on the Higgs and Drell–Yan total production cross-sections. We find that the  $$\hbox {aN}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>aN</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </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.

Alessandro Candido

Stefano Carrazza

Luigi Del Debbio

Tommaso Giani

Zahari Kassabov

Giacomo Magni

Germany

Christopher Schwan

Maria Ubiali

The path to  $$\hbox {N}^3\hbox {LO}$$  <math> <mrow> <msup> <mtext>N</mtext> <mn>3</mn> </msup> <mtext>LO</mtext> </mrow> </math>   parton distributions

We construct a set of parton distribution functions (PDFs), based on the recent NNPDF4.0 PDF set, that also include a photon PDF. The photon PDF is constructed using the LuxQED formalism, while QED evolution accounting for  $${\mathcal {O}}\left( \alpha \right) $$  <math> <mrow> <mi>O</mi> <mfenced> <mi>α</mi> </mfenced> </mrow> </math>  ,  $${\mathcal {O}}\left( \alpha \alpha _s\right) $$  <math> <mrow> <mi>O</mi> <mfenced> <mi>α</mi> <msub> <mi>α</mi> <mi>s</mi> </msub> </mfenced> </mrow> </math>  , and  $${\mathcal {O}}\left( \alpha ^2\right) $$  <math> <mrow> <mi>O</mi> <mfenced> <msup> <mi>α</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math>   corrections is implemented and benchmarked by means of the EKO code. We investigate the impact of QED effects on NNPDF4.0, and compare our results both to our previous NNPDF3.1QED PDF set and to other recent PDF sets that include the photon. We assess the impact of photon-initiated processes and electroweak corrections on a variety of representative LHC processes, and find that they can reach the 5% level in vector boson pair production at large invariant mass.

Photons in the proton: implications for the LHC

We include uncertainties due to missing higher order corrections to QCD computations (MHOU) used in the determination of parton distributions (PDFs) in the recent NNPDF4.0 set of PDFs. We use our previously published methodology, based on the treatment of MHOUs and their full correlations through a theory covariance matrix determined by scale variation, now fully incorporated in the new NNPDF theory pipeline. We assess the impact of the inclusion of MHOUs on the NNPDF4.0 central values and uncertainties, and specifically show that they lead to improved consistency of the PDF determination. PDF uncertainties on physical predictions in the data region are consequently either unchanged or moderately reduced by the inclusion of MHOUs.

Determination of the theory uncertainties from missing higher orders on NNLO parton distributions with percent accuracy

We provide a first quantitative indication that the wave function of the proton contains unequal distributions of charm quarks and antiquarks, i.e., a nonvanishing intrinsic valence charm distribution. A significant nonvanishing valence component cannot be perturbatively generated; hence, our results reinforce previous evidence that the proton contains an intrinsic (i.e., not radiatively generated) charm quark component. We establish our result through a determination of the parton distribution functions of charm quarks and antiquarks in the proton. We propose two novel experimental probes of this intrinsic charm valence component: <math><mi>D</mi></math>-meson asymmetries in <math><mrow><mi>Z</mi><mo>+</mo><mi>c</mi></mrow></math>-jet production at the LHCb experiment and flavor-tagged structure functions at the Electron-Ion Collider.

Richard D. Ball

Emanuele R. Nocera

Intrinsic charm quark valence distribution of the proton

We show that the negative polarized gluon distribution <math><mi>Δ</mi><mi>g</mi></math> found in a recent global next-to-leading-order QCD analysis of the nucleon helicity structure is incompatible with the fundamental requirement that physical cross sections must not be negative. Specifically, we show that the fact that this polarized gluon strongly violates the positivity condition <math><mo>|</mo><mi>Δ</mi><mi>g</mi><mo>|</mo><mo>≤</mo><mi>g</mi></math> in terms of the unpolarized gluon distribution <math><mi>g</mi></math> leads to negative cross sections for Higgs boson production at RHIC as a physical process, implying that this negative <math><mi>Δ</mi><mi>g</mi></math> is unphysical.

Argentina

Daniel de Florian

Werner Vogelsang

Higgs production at RHIC and the positivity of the gluon helicity distribution

We revisit our argument that shows that parton distribution functions (PDFs) in the  $$\overline{\textrm{MS}}$$  <math> <mover> <mtext>MS</mtext> <mo>¯</mo> </mover> </math>   scheme are non-negative in the perturbative region, with the dual goals of clarifying its theoretical underpinnings and elucidating its domain of validity. We specifically discuss recent results proving that PDFs can turn negative at sufficiently low scale, we clarify quantitatively various aspects of our derivation of positivity in the perturbative region, and we provide an estimate for the scale above which PDF positivity holds.

On the positivity of  $$\overline{\textrm{MS}}$$  <math> <mover> <mtext>MS</mtext> <mo>¯</mo> </mover> </math>   parton distributions

We discuss the sensitivity of theoretical predictions of observables used in searches for new physics to parton distributions (PDFs) at large momentum fraction x. Specifically, we consider the neutral-current Drell–Yan production of gauge bosons with invariant masses in the TeV range, for which the forward–backward asymmetry of charged leptons from the decay of the gauge boson in its rest frame is a traditional probe of new physics. We show that the qualitative behaviour of the asymmetry depends strongly on the assumptions made in determining the underlying PDFs. We discuss and compare the large-x behaviour of various different PDF sets, and find that they differ significantly. Consequently, the shape of the asymmetry observed at lower dilepton invariant masses, where all PDF sets are in reasonable agreement because of the presence of experimental constraints, is not necessarily reproduced at large masses where the PDFs are mostly unconstrained by data. It follows that the shape of the asymmetry at high masses may depend on assumptions made in the PDF parametrization, and thus deviations from the traditionally expected behaviour cannot be taken as a reliable indication of new physics. We demonstrate that forward–backward asymmetry measurements could help in constraining PDFs at large x and discuss the accuracy that would be required to disentangle the effects of new physics from uncertainties in the PDFs in this region.

Parton distributions and new physics searches: the Drell–Yan forward–backward asymmetry as a case study

We present a new set of parton distribution functions (PDFs) based on a fully global dataset and machine learning techniques: NNPDF4.0. We expand the NNPDF3.1 determination with 44 new datasets, mostly from the LHC. We derive a novel methodology through hyperparameter optimization, leading to an efficient fitting algorithm built upon stochastic gradient descent. We use NNLO QCD calculations and account for NLO electroweak corrections and nuclear uncertainties. Theoretical improvements in the PDF description include a systematic implementation of positivity constraints and integrability of sum rules. We validate our methodology by means of closure tests and “future tests” (i.e. tests of backward and forward data compatibility), and assess its stability, specifically upon changes of PDF parametrization basis. We study the internal compatibility of our dataset, and investigate the dependence of results both upon the choice of input dataset and of fitting methodology. We perform a first study of the phenomenological implications of NNPDF4.0 on representative LHC processes. The software framework used to produce NNPDF4.0 is made available as an open-source package together with documentation and examples.