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arxiv: 2604.13157 · v1 · submitted 2026-04-14 · ✦ hep-ph · hep-ex· nucl-ex· nucl-th

Recognition: unknown

Proton Structure from Neural Simulation-Based Inference at the LHC

Ali Kaan G\"uven, Ang Li, Claudius Krause, Daohan Wang, Elie Hammou, Jaco ter Hoeve, Juan Rojo, Lisa Benato, Luca Mantani, Maria Ubiali, Ricardo Barru\'e, Robert Sch\"ofbeck, Sergio S\'anchez Cruz

Pith reviewed 2026-05-10 15:14 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-exnucl-th
keywords proton PDFsneural simulation-based inferencetop quark pair productionLHCunbinned datagluon distributionmachine learningPDF determination
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The pith

Neural simulation-based inference extracts the proton gluon PDF from unbinned LHC data with higher precision than binned fits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper demonstrates that neural simulation-based inference can constrain the proton's parton distribution functions directly from high-dimensional unbinned detector-level data in top quark pair production at 13 TeV. Traditional PDF determinations rely on binned, low-dimensional unfolded cross sections, which discard statistical information and require approximate uncertainty treatments. The NSBI approach works with full event features while incorporating experimental and theoretical systematics, yielding tighter constraints on the gluon PDF. A reader cares because more precise PDFs underpin predictions for essentially every process measured at the LHC and its high-luminosity upgrade.

Core claim

A neural simulation-based inference pipeline applied to simulated detector-level events from top-quark pair production determines the gluon PDF with improved precision over existing low-dimensional binned analyses, while propagating both experimental and theoretical systematic uncertainties through the unbinned observables.

What carries the argument

Neural simulation-based inference pipeline that ingests unbinned high-dimensional detector-level features and returns posterior constraints on PDF parameters, trained on simulations that include all modeled uncertainties.

If this is right

  • Information loss from binning is eliminated, preserving the full statistical power of the data.
  • Systematic uncertainties and their correlations are handled directly without coarse approximations.
  • The method supports a shift toward unbinned detector-level machine-learning-assisted measurements at the LHC.
  • Tighter PDF constraints improve theoretical predictions for all processes, especially at high luminosity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same unbinned pipeline could be applied to other final states to constrain additional PDF flavors.
  • Integration into global fits would combine this unbinned information with existing datasets.
  • Real-data deployment would still require extensive validation against known PDF benchmarks.
  • The approach may reduce dependence on separate unfolding procedures in future analyses.

Load-bearing premise

The Monte Carlo simulations faithfully reproduce the statistical and systematic structure of real LHC data; any mismatch biases the inferred PDFs.

What would settle it

Running the trained NSBI pipeline on actual LHC top-pair collision data and comparing the resulting gluon PDF uncertainty bands to those from standard global binned fits would show whether the precision gain materializes.

Figures

Figures reproduced from arXiv: 2604.13157 by Ali Kaan G\"uven, Ang Li, Claudius Krause, Daohan Wang, Elie Hammou, Jaco ter Hoeve, Juan Rojo, Lisa Benato, Luca Mantani, Maria Ubiali, Ricardo Barru\'e, Robert Sch\"ofbeck, Sergio S\'anchez Cruz.

Figure 2.1
Figure 2.1. Figure 2.1: The first N = 8 eigenvectors xφ (a) g (x, Q0) for the linear model of the gluon PDF (top) obtained with the POD procedure for the reference scale Q0 = 1.65 GeV for the ranges 2 · 10−5 ≤ x ≤ 1 (left) and 10−2 ≤ x ≤ 1 (right). Quark PDFs are set to be equal to the central value of PDF4LHC21. The same eigenvectors are also shown for Q = 1 TeV (middle), highlighting the effects of DGLAP evolution on the basi… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The numerical value of the homogeneous momentum integral, Eq. (2.10), Eq. (2.25), which should be satisfied by the basis functions of our linear model for the gluon PDF at all scales Q ≥ Q0. We evaluate it for the first N = 30 eigenvectors of the model at three different scales: Q = 1.65 GeV, Q = 100 GeV, and Q = 10 TeV. Note that for Q = Q0 = 1.65 GeV φ (a) Σ (x) = 0 for all a. The quark PDFs have been … view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The reconstruction quality of PDF4LHC21 gluon replicas is quantified as the distance to the gluon PDF targets d (T) rec,j (N, Q), Eq. (2.27), evaluated for the NT = 100 original and reconstructed replicas of the PDF4LHC21 NNLO set and as a function of the POD basis dimension N. The central values (dashed curves) indicate the median distance DT (Q), and the associated band indicates the corresponding stan… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The 68% CL intervals on the coefficients ca with a = 1, . . . , 9 for the linear PDF model fitted to the same PDF4LHC21 replicas as in [PITH_FULL_IMAGE:figures/full_fig_p011_2_4.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The distribution of x and µF of gluon-initiated sub-processes of the tt(2ℓ) signal sample, evaluated with POWHEG and NNPDF3.1 in the acceptance region (left). The contours indicate the region containing 68%, 95%, and 99% of the events in the sample. One-dimensional distribution of x for all partonic sub-processes (right). 500 1000 1500 2000 2500 3000 1 10 2 10 3 10 Number of events gg qg qq 500 1000 1500… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: One-dimensional distribution of m(tt) for the partonic channels gg, qg, and qq¯ (left). The quantiles of the distribution of x in bins of m(tt), separately for the gg (middle) and qg (right) partonic channels. For qg, we separate the gluon leg and the quark leg. Statistical uncertainties are shown as shaded bands. can be safely neglected. The left panel of [PITH_FULL_IMAGE:figures/full_fig_p017_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The relative difference between the PDF weight evaluated by POWHEG’s native reweighting algorithm and the offline evaluation based on Eq. (4.4) for each of the Neig = 100 eigenvectors of NNPDF3.1 (Hessian variant) for a specific Monte Carlo event from our sample. Qualitatively similar results are obtained for other events. to ≈ 0.25 at the 95% CL, while for quark-gluon scattering the quark leg reaches x … view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Distributions of detector-level training features in our event sample (black histograms) and the total systematic uncertainty (shaded bands) associated to them: invariant mass and pT of the top quark pair (top); rapidity of the top quark pair (center left); difference of η of the tt system (center right); difference of |η| of the tt system (bottom left); pT of the top quark (bottom right). The colored cu… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Distributions of detector-level training features in our event sample (black histograms) and the total systematic uncertainty (shaded bands) associated to them: pT of the anti-top quark (top left); rapidity of the top quark (top right) and of the anti-top quark (center left); pT of the leading lepton (center right) and of subleading lepton (bottom left); pT of the ℓℓ system (bottom right). The colored cu… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Distributions of detector-level training features in our event sample (black histograms) and the total systematic uncertainty (shaded bands) associated to them: difference of η of the ℓℓ system (top left); difference of |η| of the ℓℓ system (top right); subleading lepton (bottom left); invariant mass and pseudorapidity of the ℓℓ system (bottom). The colored curves indicate the relative variations of thes… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Validation of the learned Rˆ(x, c). We show, as a representative example, the distribution of Rˆ 1(x) (top panel) and the associated residual (bottom panel). The uncertainty band corresponds to the 1σ spread of the residual and reflects the latent uncertainty. In the left panel of [PITH_FULL_IMAGE:figures/full_fig_p025_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: The gluon PDF xfg(x, Q0) in the linear model for the first 6 eigenvectors from the PCA of the expected unbinned Fisher information matrix Eq. (4.9) evaluated for c = v (k) (left). The bottom panel displays the ratio to the central gluon of the linear model. The mode decomposition fraction v (k)2 a for the same eigenvectors (right). The y-axis indicates the corresponding eigenvalue. All eigenvectors recei… view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: The gluon PDF xfg(x, Q0) in the linear model for the first 6 eigenvectors from the PCA of the expected unbinned Fisher information matrix Eq. (4.9) evaluated for the normalised eigen-directions, c = ±w(k) . The lower panel shows the ratio to the reference PDF. When evaluated along these normalised eigen-directions of the Fisher information matrix, the variations of the linear model are clearly smaller in… view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Validation of the learned MHOU surrogate Sˆmhou(x, νR, νF ). The variations of the reconstructed m(tt) distribution (markers) for the shifts in (νR, νF ) listed in Eq. (4.20), compared with the corresponding surrogate predictions shown as lines (left). The C2ST (right), performed for (νR, νF ) = (1, 1), comparing the nominal sample with the shifted sample before reweighting (orange dashed) and after rew… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: The expected precision of the gluon PDF from top quark pair production at the LHC, comparing the outcome of the binned and NSBI unbinned analyses at both Q0 = 1.65 GeV (top) and Q = 175 GeV (bottom panels), for a logarithmic (left) and linear (right panels) scale. Results are shown normalized to the central value of the reference gluon PDF, and we indicate the 68% CL uncertainties in each case. computed.… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Same as [PITH_FULL_IMAGE:figures/full_fig_p035_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Comparison of the binned and NSBI unbinned uncertainties to the symmetrized uncertainty envelope of NNPDF3.1, NNPDF4.0, CT18, and MSHT20 for Q = 1.65 GeV (top) and Q = 175 GeV (bottom), on a logarith￾mic (left) and linear (right) scale. of more elements to the basis functions, provided that quasi-flat directions with low eigenvalues after PCA rotation are weeded out. Without this removal, increasing the … view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Comparison of the binned and NSBI unbinned uncertainties for linear models based on N = 6 and N = 7 elements, respectively, and for Q = 1.65 GeV (left) and Q = 175 GeV (right). The numbers in parenthesis indicate the number of POIs after PCA rotation and the removal of quasi-flat directions. 10−2 10−1 x 0.6 0.8 1.0 1.2 1.4 δg/g(x, Q2) Q = 1.65 GeV Binned (stat only) Binned (stat + syst) NSBI unbinned (st… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Comparison of the binned and NSBI unbinned uncertainties for N = 6 without systematic uncertain￾ties (dashed) to the nominal result (solid), shown for Q = 1.65 GeV (left) and Q = 175 GeV (right). values of the linear model parameters are consistent with zero, as expected for a successful closure test. Here we demonstrate that also when the Asimov data are generated with a different hypothesis for the glu… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Fit of the linear gluon PDF model to toy data sampled from a simulated data set (top left) according to the central element of PDF4LHC21 (f (g) target). The mean of the toy distribution (solid black) is in good agreement with the target gluon PDF (dashed green). The distribution of the negative log-likelihood (top right) and of the first two principal components (bottom) in Eq. (4.11) are also shown. (yh… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Double-differential cross section of Higgs production in gluon fusion, evaluated at NLO in QCD with Madgraph5 aMC@NLO as a function of absolute rapidity |yh| and transverse momentum p h T of the Higgs boson. PDF uncertainties are evaluated for NNPDF3.1 NNLO (orange) and for binned (green) and NSBI unbinned (blue) determinations of the gluon PDF presented in this work. gluon-fusion Higgs production based … view at source ↗
read the original abstract

The precise determination of the parton distribution functions (PDFs) of the proton is an essential ingredient for LHC analyses, including for those at the upcoming High-Luminosity LHC. So far, PDFs are determined from global fits to binned low-dimensional data obtained from unfolded hard-scattering cross section measurements. In this work we demonstrate for the first time the feasibility of neural simulation-based inference (NSBI) for constraining the proton PDFs using a high-dimensional unbinned data set. Exploiting the full statistical power of unbinned data removes the loss of information inherited by the binning procedure. As a proof-of-concept, we determine the gluon PDF from simulated data of top quark pair production at the LHC with $\sqrt{s}=13$ TeV. Taking into account both experimental and theoretical systematic uncertainties in the detector-level features, we demonstrate how the NSBI pipeline achieves significant improvements in precision compared to existing low-dimensional binned analyses. Our results illustrate the potential of unbinned inference to reduce the reliance on coarse approximations of uncertainties and their correlations entering PDF determinations, hence contributing to a new paradigm of unbinned detector-level ML-assisted measurements at the LHC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a proof-of-concept for neural simulation-based inference (NSBI) to constrain the gluon PDF from high-dimensional unbinned simulated detector-level data in top-quark pair production at 13 TeV. It incorporates experimental and theoretical systematic uncertainties and claims significant precision gains over traditional low-dimensional binned PDF fits.

Significance. If substantiated, the approach could enable a shift toward unbinned, ML-assisted PDF extractions that retain more statistical information and model uncertainties more directly. The controlled simulation setting is appropriate for demonstrating feasibility and highlights potential for reduced reliance on binning approximations in future LHC analyses.

major comments (1)
  1. [Results] The central claim of 'significant improvements in precision' relative to binned analyses is not accompanied by quantitative metrics, closure tests, or comparison tables. This information is required to evaluate whether the NSBI pipeline delivers the asserted gains on the simulated dataset.
minor comments (1)
  1. [Abstract] The abstract would benefit from a brief quantitative indication of the precision improvement to allow readers to gauge the result immediately.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and the positive assessment of the potential impact of our work. We address the major comment below and will revise the manuscript to strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [Results] The central claim of 'significant improvements in precision' relative to binned analyses is not accompanied by quantitative metrics, closure tests, or comparison tables. This information is required to evaluate whether the NSBI pipeline delivers the asserted gains on the simulated dataset.

    Authors: We agree that the manuscript would be strengthened by the inclusion of explicit quantitative metrics, closure tests, and direct comparison tables. While the current version demonstrates the NSBI results through uncertainty bands on the gluon PDF (as shown in the relevant figures), we acknowledge that these do not provide the side-by-side numerical comparisons needed for a rigorous evaluation. In the revised manuscript we will add a dedicated subsection with closure tests on the simulated dataset (injecting known PDF variations and recovering them) and a table reporting the relative reduction in gluon PDF uncertainties at representative x values between the NSBI approach and the traditional binned fit on identical data. This will make the precision gains fully transparent and quantifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; minor self-citation not load-bearing

full rationale

The paper presents a feasibility study of NSBI applied to simulated detector-level top-pair events to constrain the gluon PDF, with explicit comparison to external binned PDF fits. No derivation step reduces a claimed prediction to a fitted parameter or self-citation by construction; the pipeline trains on forward simulation whose generative model is known by design, and results are validated against independent binned analyses. Self-citations to prior NSBI or PDF work exist but are not invoked as uniqueness theorems or ansatze that force the central result. The simulation-fidelity assumption is a scope limitation, not a circularity in the inference chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The method implicitly relies on neural-network hyperparameters and the fidelity of the Monte Carlo simulation chain.

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discussion (0)

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Reference graph

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