Proton Structure from Neural Simulation-Based Inference at the LHC

This paper demonstrates for the first time that neural simulation-based inference (NSBI) can effectively constrain proton parton distribution functions using high-dimensional, unbinned LHC data, achieving significantly improved precision over traditional binned analyses by fully exploiting statistical power and reducing reliance on coarse uncertainty approximations.

The Gist

The Big Picture: Decoding the Proton's "Recipe"

Imagine the proton (the particle inside an atom's nucleus) not as a solid ball, but as a busy, chaotic kitchen. Inside this kitchen, there are ingredients called quarks and gluons (the "glue" holding them together).

To understand how the universe works, physicists need to know the exact "recipe" of this kitchen: How much glue is there? Is it spread out evenly, or clumped in corners? This recipe is called the Parton Distribution Function (PDF).

Currently, we have a good idea of the recipe, but it's like trying to guess a cake's recipe by only tasting a few crumbs from the edge. We need to taste the whole cake to be sure.

The Problem: The "Blender" Effect

For decades, scientists have studied protons by smashing them together at the Large Hadron Collider (LHC). However, the way they analyzed the data was like taking a high-definition video of the crash and blending it into a smoothie.

• The Old Way (Binned Data): They would group all the crash results into big buckets (bins). "How many crashes happened in this speed range? How many in that one?"
• The Flaw: When you blend a smoothie, you lose the texture. You lose the specific details of individual particles. You also have to guess how the different ingredients might have influenced each other (correlations). This "blending" throws away precious information and makes the recipe less precise.

The Solution: Neural Simulation-Based Inference (NSBI)

This paper introduces a new method called NSBI. Think of it as switching from a blender to a super-smart AI chef.

Instead of grouping data into buckets, the AI looks at every single particle from every single crash individually. It's like looking at the video frame-by-frame, noticing the exact spin of every crumb, and using that to reconstruct the recipe with incredible precision.

How They Did It: The "Top Quark" Test

To prove this new method works, the scientists used a specific type of crash: Top Quark Pair Production.

• The Analogy: Imagine the Top Quark is a very heavy, rare fruit that only grows in a specific part of the proton's kitchen. Because it's so heavy, it requires a lot of "glue" (gluons) to create it.
• The Experiment: They simulated millions of these crashes using a computer. They didn't just count the fruits; they looked at the exact angle, speed, and energy of every piece of debris flying out.

The Magic Trick: The "Linear Model"

The math behind the proton is incredibly complex. To make it manageable for the AI, the authors built a Linear Model.

• The Analogy: Imagine the proton's recipe is a giant, complex song. Instead of trying to learn the whole song at once, they broke it down into 8 simple musical notes (basis functions).
• The AI's job wasn't to learn the whole song; it just had to figure out how loud to play each of the 8 notes to match the data. This made the math much faster and more stable.

The Results: Sharper Vision

When they compared the new "AI Chef" (Unbinned NSBI) against the old "Blender" (Binned analysis), the results were stunning:

1. Precision: The new method was significantly more precise. It reduced the uncertainty in the "glue" (gluon) recipe by a large margin.
2. Systematic Errors: In the old method, small errors in the detector (like a slightly miscalibrated scale) could ruin the whole recipe. The new AI method is much better at ignoring these small errors and focusing on the true signal.
3. Independence: Usually, to get a good recipe, you need data from many different experiments (like tasting the cake, the frosting, and the plate). This new method suggests that one single experiment (Top Quark crashes) might be enough to get a perfect recipe for the gluon, without needing help from outside data.

Why This Matters for the Future

The LHC is about to get a massive upgrade called the High-Luminosity LHC, which will produce 10 times more data.

• The Challenge: If we keep using the "blender" method, we will just have a smoothie with 10 times more noise. We will still be guessing.
• The Future: This paper shows that by using Unbinned NSBI, we can use that massive amount of data to finally see the proton's structure in high definition.

In short: This paper proves that by using advanced AI to look at every single particle individually—instead of grouping them into buckets—we can finally write down the perfect recipe for the proton, leading to a deeper understanding of the universe.

Technical Summary

1. Problem Statement

The precise determination of Parton Distribution Functions (PDFs), particularly the gluon PDF, is critical for Standard Model predictions and New Physics searches at the Large Hadron Collider (LHC). Current global PDF fits rely on binned, low-dimensional observables (e.g., differential cross-sections in histograms) derived from unfolded experimental data. This approach suffers from three major bottlenecks:

1. Information Loss: Binning integrates out high-dimensional statistical information, reducing sensitivity.
2. Approximation Errors: Experimental uncertainties are often approximated as multivariate Gaussians, which fails near kinematic edges and conflates systematic correlations.
3. Systematic Limitations: The reliance on coarse binning limits the ability to constrain systematic uncertainties directly from the data distribution.

The paper addresses the challenge of performing unbinned, high-dimensional inference on detector-level data to extract PDFs while rigorously accounting for both experimental and theoretical systematic uncertainties.

2. Methodology

The authors propose a novel framework combining Neural Simulation-Based Inference (NSBI) with a linear model for the gluon PDF. The methodology consists of four main pillars:

A. Linear Model for the Gluon PDF

Instead of using complex neural networks to parameterize the PDF directly, the authors construct a linear model based on Proper Orthogonal Decomposition (POD).

• Construction: A Hilbert space of physically admissible gluon PDFs is generated using deep neural networks (NNPDF architecture). This space is reduced to a finite-dimensional basis ($N$ basis functions) via POD.
• Formalism: The gluon PDF is expressed as $f_g(x, Q, c) = \phi^{(0)}_g(x, Q) + \sum_{a=1}^N c_a \phi^{(a)}_g(x, Q)$, where $c_a$ are the parameters of interest (POIs).
• Constraints: The model strictly enforces theoretical constraints (momentum sum rules, positivity, integrability) and DGLAP evolution.
• Quadratic Dependence: Because the hadronic cross-section involves the product of two PDFs, the dependence on the coefficients $c$ is at most quadratic. This analytic structure is crucial for the efficiency of the ML surrogates.

B. Neural Simulation-Based Inference (NSBI)

The core of the analysis uses NSBI to infer the coefficients $c$ directly from unbinned, detector-level event features ($x$).

• Surrogate Modeling: Since the detector response is intractable, the authors train machine learning surrogates to approximate the likelihood ratio.
  • PDF Dependence ($R(x, c)$): Modeled as an exact quadratic polynomial in $c$, learned using a Boosted Information Tree (BIT) algorithm. This algorithm learns the full polynomial expansion simultaneously rather than coefficient-by-coefficient.
  • Systematic Dependence ($S(x, \nu)$): Systematic uncertainties (nuisance parameters $\nu$) are modeled using parametric neural networks that learn the log-polynomial dependence on $\nu$.
• Likelihood Construction: An extended likelihood is constructed using the ratio of the surrogate predictions to the nominal simulation. The analysis utilizes the profiled likelihood-ratio test statistic to constrain $c$ while profiling over nuisance parameters.

C. Data and Systematics

• Process: Top-quark pair production ($t\bar{t}$) in the dileptonic channel ($pp \to t\bar{t} \to b\ell^+\nu b\ell^-\bar{\nu}$) at $\sqrt{s} = 13$ TeV.
• Observables: The analysis uses 16 unbinned event-level features (kinematics of top quarks, leptons, and the $t\bar{t}$ system) rather than binned histograms.
• Systematics: A comprehensive set of uncertainties is included, such as Jet Energy Scale (JES), Jet Energy Resolution (JER), $b$-tagging efficiencies, lepton efficiencies, and theoretical uncertainties (Missing Higher Order Uncertainties via scale variations and $\alpha_s$ variations). These are learned via ML surrogates trained on simulated variations.

D. Principal Component Analysis (PCA)

To handle numerical instabilities caused by "quasi-flat" directions in the parameter space (combinations of $c_a$ that the data cannot constrain), the authors perform PCA on the Fisher Information Matrix. This identifies the optimal linear combinations of basis coefficients that are actually constrained by the $t\bar{t}$ data, effectively reducing the dimensionality of the fit.

3. Key Contributions

1. First Unbinned PDF Determination: This is the first demonstration of constraining proton PDFs using unbinned, high-dimensional detector-level data via NSBI, bypassing the information loss inherent in binning.
2. Integrated Systematics Handling: The framework successfully integrates complex experimental and theoretical systematic uncertainties directly into the inference pipeline using ML surrogates, avoiding the need for coarse Gaussian approximations of correlations.
3. Linear Model Validation: The authors validate that a linear model with $N=6$ basis functions is sufficient to reconstruct the gluon PDF with high accuracy (mismatch < 10% of native PDF uncertainty) in the relevant kinematic region ($10^{-2} \le x \le 0.5$).
4. Boosted Information Tree (BIT): The paper extends the BIT algorithm to learn full polynomial dependencies (quadratic in this case) in a single boosting sequence, improving efficiency and stability.

4. Results

The study uses Asimov datasets (simulated data generated from the central PDF) to assess expected precision:

• Precision Improvement: The unbinned NSBI analysis achieves significantly higher precision than a reference binned analysis (using the same 16 observables but binned into a $4 \times 4$ grid).
  • At $Q = 175$ GeV, the unbinned method reduces uncertainties by a factor of $\sim 2$ in the region $x \sim 0.1$ compared to the binned approach when systematics are included.
• Comparison to Global Fits: The precision of the gluon PDF extracted from a single $t\bar{t}$ unbinned measurement is competitive with, and in some regions superior to, global PDF fits (like NNPDF4.0, CT18, MSHT20) that combine dozens of different datasets.
  • Specifically, for $0.01 \lesssim x \lesssim 0.35$, the NSBI unbinned uncertainty matches the best global fits.
• Systematics Robustness: The unbinned method is shown to be more efficient at constraining systematic nuisance parameters from the data itself, whereas the binned method suffers a larger degradation in precision when systematics are included.
• Higgs Production Implications: Applying the derived gluon PDF to Higgs boson production in gluon fusion ($gg \to H$) shows that theoretical predictions can achieve precision comparable to global fits, suggesting that LHC experiments could perform internal PDF calibration without relying on external datasets (like HERA).

5. Significance and Outlook

• New Paradigm for LHC: The work proposes a shift from "binned, unfolded" analyses to "unbinned, detector-level" analyses. This is essential for the High-Luminosity LHC (HL-LHC) era, where statistical power is high, and preserving information is critical.
• Internal Calibration: It demonstrates that LHC experiments can determine proton structure inputs using their own data (e.g., $t\bar{t}$), reducing reliance on external constraints and allowing for better control of correlated systematic errors.
• Future Applications: The methodology is generalizable to other processes (Drell-Yan, single-top) and other parameters (SMEFT Wilson coefficients, $\alpha_s$, top mass). It paves the way for combined fits of PDFs and BSM parameters using high-dimensional data.
• Next Steps: The authors plan to apply this method to real LHC data, requiring NNLO-accurate event generators (e.g., MiNNLOPS) for theoretical predictions and the release of unbinned likelihoods by experimental collaborations.

In conclusion, this paper establishes Neural Simulation-Based Inference as a viable and superior alternative to traditional binned methods for PDF determination, offering a path to significantly reduced uncertainties in proton structure measurements at the LHC.