This work provides a proof of concept for the computation of pure gluonic amplitudes in quantum chromodynamics (QCD) on graphics processing units (GPUs). The implementation relies on the Berends–Giele recursion algorithm and, for the first time on a GPU, enables the numerical computation of amplitudes in an arbitrary number of space-time dimensions and over finite fields. This demonstrates the advantages of hardware acceleration, not only for the computation of tree-level amplitudes for real-radiation processes in four dimensions over complex numbers but also for generating loop integrands for virtual corrections in $\textit{d}$ dimensions over finite fields. The associated computer program is publicly available.

Switzerland

Juan Cruz-Martinez

United Kingdom

Giuseppe De Laurentis

Germany

Mathieu Pellen

Accelerating Berends–Giele recursion for gluons in arbitrary dimensions over finite fields

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

Italy

Stefano Forte

Niccolò Laurenti

Netherlands

Tanjona Rabemananjara

Juan Rojo

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.

Richard Ball

Andrea Barontini

Alessandro Candido

Stefano Carrazza

Luigi Del Debbio

Tommaso Giani

Finland

Felix Hekhorn

Zahari Kassabov

Giacomo Magni

Emanuele Nocera

Christopher Schwan

Roy Stegeman

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

Proton-proton collisions at the LHC generate a high-intensity collimated beam of neutrinos in the forward (beam) direction, characterised by energies of up to several TeV. The recent observation of LHC neutrinos by FASER $$\nu $$  <math> <mi>ν</mi> </math>   and SND@LHC signifies that this previously overlooked particle beam is now available for scientific investigation. Here we quantify the impact that neutrino deep-inelastic scattering (DIS) measurements at the LHC would have on the parton distributions (PDFs) of protons and heavy nuclei. We generate projections for DIS structure functions for FASER $$\nu $$  <math> <mi>ν</mi> </math>   and SND@LHC at Run III, as well as for the FASER $$\nu $$  <math> <mi>ν</mi> </math>  2, AdvSND, and FLArE experiments to be hosted at the proposed Forward Physics Facility (FPF) operating concurrently with the High-Luminosity LHC (HL-LHC). We determine that up to one million electron-neutrino and muon-neutrino DIS interactions within detector acceptance can be expected by the end of the HL-LHC, covering a kinematic region in x and  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   overlapping with that of the Electron-Ion Collider. Including these DIS projections in global (n)PDF analyses, specifically PDF4LHC21, NNPDF4.0, and EPPS21, reveals a significant reduction in PDF uncertainties, in particular for strangeness and the up and down valence PDFs. We show that LHC neutrino data enable improved theoretical predictions for core processes at the HL-LHC, such as Higgs and weak gauge boson production. Our analysis demonstrates that exploiting the LHC neutrino beam effectively provides CERN with a “Neutrino-Ion Collider” without requiring modifications in its accelerator infrastructure.

United States

Max Fieg

Peter Krack

Poland

Toni Mäkelä

The LHC as a Neutrino-Ion Collider

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.

Shayan Iranipour

United Arab Emirates

Singapore

Spain

Jose Latorre

Rosalyn Pearson

Cameron Voisey

Michael Wilson

The path to proton structure at 1% accuracy

Since the first determination of a structure function many decades ago, all methodologies used to determine structure functions or parton distribution functions (PDFs) have employed a common prefactor as part of the parametrization. The NNPDF collaboration pioneered the use of neural networks to overcome the inherent bias of constraining the space of solution with a fixed functional form while still keeping the same common prefactor as a preprocessing. Over the years various, increasingly sophisticated, techniques have been introduced to counter the effect of the prefactor on the PDF determination. In this paper we present a methodology to perform a data-based scaling of the Bjorken x input parameter which facilitates the removal the prefactor, thereby significantly simplifying the methodology, without a loss of efficiency and finding good agreement with previous results.

A data-based parametrization of parton distribution functions

We present the software framework underlying the NNPDF4.0 global determination of parton distribution functions (PDFs). The code is released under an open source licence and is accompanied by extensive documentation and examples. The code base is composed by a PDF fitting package, tools to handle experimental data and to efficiently compare it to theoretical predictions, and a versatile analysis framework. In addition to ensuring the reproducibility of the NNPDF4.0 (and subsequent) determination, the public release of the NNPDF fitting framework enables a number of phenomenological applications and the production of PDF fits under user-defined data and theory assumptions.

An open-source machine learning framework for global analyses of parton distributions