AI-assisted modeling and Bayesian inference of unpolarized quark transverse momentum distributions from Drell-Yan data

This paper presents a global Bayesian analysis of unpolarized quark transverse-momentum-dependent parton distribution functions using Drell-Yan data at ${\rm N^3LO}$ and ${\rm N^4LL}$ accuracy, leveraging AI-driven functional form selection and machine-learning emulators to enable efficient Markov Chain Monte Carlo sampling and quantify uncertainties.

The Gist

The Big Picture: Mapping the Invisible Proton

Imagine the proton (the tiny particle inside an atom's nucleus) not as a solid marble, but as a busy, chaotic city filled with smaller particles called quarks and gluons.

For a long time, physicists have had a very good map of this city showing how the traffic moves forward and backward (longitudinal motion). But they were missing a crucial part of the map: how the traffic swerves side-to-side (transverse motion). This side-to-side movement is called the "Transverse Momentum."

This paper is about creating a high-definition, 3D map of that side-to-side swerving. The authors used data from massive particle collisions (like the Drell-Yan process, where particles smash together and create pairs of leptons) to figure out exactly how these quarks move.

The New Toolkit: AI as the Architect and the Speedster

The authors didn't just crunch numbers; they used Artificial Intelligence (AI) in two clever ways to solve a problem that is usually too hard for humans to do alone.

1. The AI Architect (Designing the Map)

Usually, when scientists try to map something they can't see directly, they have to guess the shape of the curve that fits the data. It's like trying to draw a smooth line through a scatter of dots on a piece of paper.

• The Old Way: A human would guess a shape (maybe a curve, maybe a wave), draw it, and see if it fits. If it doesn't, they try a different shape. This is slow and biased by what the human thinks the shape should look like.
• The New Way (This Paper): The authors used an AI agent as an Architect. They told the AI, "Here are the rules of physics (the building codes). Now, go design 100 different blueprints for the curve that fits our data best." The AI generated, tested, and ranked hundreds of mathematical shapes automatically. It found the best design much faster and more creatively than a human could, ensuring they didn't miss a better shape just because it looked "weird."

2. The AI Speedster (The Emulator)

Once they had the best design, they needed to test it against thousands of data points.

• The Problem: Calculating the physics for every single data point is like trying to bake a cake from scratch every time you want to check if the frosting is right. It takes forever. If you need to check it a million times (which you do in statistics), you'd be baking for years.
• The Solution: They trained a Machine Learning Emulator. Think of this as a super-fast food critic who has tasted a million cakes. Instead of baking a new cake (doing the full physics calculation), the critic looks at the ingredients and instantly says, "This will taste like a 9.5/10."
• This "emulator" learned the relationship between the inputs and the results so well that it could predict the outcome in a split second. This allowed the scientists to run their complex statistical tests in days instead of years.

The Detective Work: Two Ways to Measure Uncertainty

The paper compares two different ways of measuring how "sure" they are about their map.

1. The "Clone Army" Method (Replica Analysis):
   Imagine you have a blurry photo of a suspect. To figure out who it is, you create 100 slightly different versions of the photo (adding random noise to the pixels) and ask 100 different detectives to identify the person. If 95 of them say "It's Bob," you are pretty sure it's Bob. This is the traditional method used in physics.

2. The "Bayesian Detective" Method (Bayesian Inference):
   This is a more modern, probabilistic approach. Instead of making clones of the photo, the detective starts with a "gut feeling" (a prior belief) about who the suspect might be. As they look at the evidence, they update their belief mathematically. It's like a detective saying, "I thought it was Bob, but the new evidence makes me 90% sure it's Bob, with a 10% chance it's Charlie."

The Result: The authors found that both methods agreed on who the suspect was (the central values of the map were the same). However, the Bayesian method gave a slightly wider range of "maybe" (larger uncertainty bands). This is actually good news! It means the Bayesian method is being more honest about the unknowns, rather than pretending to be more certain than the data allows.

The Findings: What Did They Learn?

• The "Swerve" is Real: They successfully mapped how quarks move side-to-side.
• The "Collins-Soper Kernel": This is a fancy name for the "engine" that drives how the quarks' movement changes as the energy of the collision changes. Their new map for this engine matches up very well with other recent discoveries (like those from the Electron-Energy Correlator) and even with supercomputer simulations (Lattice QCD).
• Consistency: Whether they used the "Clone Army" or the "Bayesian Detective," the final picture of the proton's interior looked very similar.

Why Does This Matter?

Think of the proton as a complex machine. If you want to build a better engine (or understand the universe's fundamental forces), you need to know exactly how every gear turns.

By using AI to design the best mathematical models and to speed up the calculations, this paper provides the most precise map yet of how quarks move sideways inside a proton. It also proves that AI and Bayesian statistics are powerful new tools that can work together to solve problems that were previously too difficult or too slow to tackle.

In short: They used AI to design the best possible map, used a digital "speedster" to draw it quickly, and compared two different ways of measuring the map's accuracy to ensure it's reliable. The result is a clearer, more honest picture of the building blocks of our universe.

Technical Summary

1. Problem Statement

The paper addresses the challenge of extracting unpolarized quark Transverse-Momentum-Dependent Parton Distribution Functions (TMD PDFs) from Drell–Yan (DY) data with reliable uncertainty quantification.

• Context: TMD PDFs describe the 3D momentum structure of the proton. While theoretical perturbative QCD (pQCD) ingredients have reached high precision (N3LO + N4LL resummation), the nonperturbative sector (large transverse distances) requires phenomenological modeling.
• The Gap: Traditional extractions rely heavily on the Monte Carlo replica method for uncertainty estimation. While robust, this method is computationally intensive and lacks the explicit probabilistic framework of Bayesian inference. Furthermore, the choice of functional forms for nonperturbative parameters is often subjective or limited by manual exploration.
• Goal: To perform a global extraction of TMD PDFs using a Bayesian inference framework, incorporating Artificial Intelligence (AI) to optimize model selection and accelerate computation, and to directly compare the resulting uncertainties with the standard replica method.

2. Methodology

The authors developed a sophisticated workflow integrating high-order pQCD, AI-driven model selection, and machine learning surrogates.

A. Theoretical Framework

• Process: Neutral-current Drell–Yan production ($h_1 h_2 \to \ell^+\ell^- X$).
• Accuracy: Calculations are performed at N3LO (Next-to-Next-to-Next-to-Leading Order) in perturbative QCD combined with N4LL (Next-to-Next-to-Next-to-Next-to-Leading Logarithmic) resummation.
• Factorization: Uses TMD factorization for the low-$q_T$ region. The cross-section involves hard functions, perturbative matching coefficients, and nonperturbative functions ($S_{NP}$ for the TMD PDF and $D_{NP}$ for the Collins–Soper kernel).
• Data: A global dataset of 465 data points from fixed-target experiments (E288, E605, E772), RHIC (STAR), Tevatron (CDF, D0), and LHC (ATLAS, CMS, LHCb) at various energies (7–13 TeV).

B. AI-Driven Model Exploration (Section 4.3)

Instead of fixing the nonperturbative functional forms a priori, the authors employed an AI agent (OpenAI Codex/GPT-5.4) to explore the "ansatz space."

• Workflow: The agent was given phenomenological goals (e.g., fit quality, physical behavior in $k_T$-space, avoiding parameter correlations) and physics constraints (e.g., $S_{NP}(b \to 0) = 1$, $D_{NP}(b \to 0) \sim b^2$).
• Process: The agent iteratively proposed, refitted, and evaluated candidate functional forms for the nonperturbative Sudakov factor $S_{NP}(x,b)$ and the Collins–Soper kernel $D_{NP}(b)$.
• Outcome: This led to the selection of a specific parameterization involving 9 free parameters (e.g., a Gaussian-like deformation in $\ln x$ for $S_{NP}$ and a specific saturation profile for $D_{NP}$), which was found to be superior to traditional forms.

C. Bayesian Inference & ML Emulator (Section 5)

Bayesian inference requires evaluating the likelihood function millions of times, which is computationally prohibitive with exact theory calculations.

• Surrogate Model: The authors trained a Multilayer Perceptron (MLP) emulator to predict the "whitened residual vector" (the difference between theory and data, normalized by the covariance matrix).
• Adaptive Training: An AI-driven "controller-executor-reviewer" workflow was used to generate training data. The controller identified "hard" regions (high error) in the parameter space, and the executor generated new truth-level data points there to retrain the emulator, ensuring high accuracy where it mattered most.
• Sampling: The posterior distribution was sampled using an affine-invariant ensemble MCMC sampler (emcee).
• Marginalization: Collinear PDF uncertainties were incorporated by marginalizing over an ensemble of 100 collinear PDF replicas within the likelihood function.

D. Comparison with Replica Method

A parallel analysis was performed using the standard Monte Carlo replica method (generating 100 pseudodata sets and refitting) to serve as a non-Bayesian control for direct comparison.

3. Key Contributions

1. First Global Bayesian TMD Extraction: This is one of the first applications of full Bayesian inference to a global TMD analysis, providing a direct probabilistic interpretation of uncertainties.
2. AI-Optimized Nonperturbative Modeling: Demonstrated that AI agents can systematically explore and rank functional forms for nonperturbative physics, reducing human bias and discovering flexible parameterizations that improve fit quality.
3. ML-Accelerated Bayesian Inference: Successfully implemented a machine learning emulator to make high-dimensional Bayesian sampling feasible for complex QCD calculations, maintaining high accuracy while drastically reducing computational cost.
4. Methodological Comparison: Provided a rigorous, quantitative comparison between Bayesian and Replica uncertainty estimates within the same theoretical framework, revealing distinct structural differences in how uncertainties are propagated.

4. Results

A. Fit Quality and Parameters

• Fit Quality: Both methods achieved excellent global fit quality, with $\chi^2/N \approx 1.02$ (Replica) and $\chi^2/N \approx 1.03$ (Bayesian).
• Parameter Values: The central values of the 9 nonperturbative parameters were generally consistent between methods, though they corresponded to slightly different local minima.
• Uncertainty Structure:
  • Bayesian Uncertainties: Generally broader than replica uncertainties (by a factor of ~1.23 on average).
  • Correlations: The correlation structures differed. For instance, the Bayesian analysis showed stronger interplay between the shape and Gaussian subsets of parameters, whereas the replica method showed a strong anti-correlation between the Collins–Soper parameters $c_1$ and $B_{NP}$.
  • Gaussian Limit: The Bayesian posterior was found to be closer to a Gaussian distribution (Hessian approximation) than the replica distribution, suggesting Bayesian results may be more robust for use in subsequent fitting pipelines relying on Gaussian statistics.

B. Physics Outputs

• Collins–Soper Kernel: The extracted kernel $D(b, \mu)$ is consistent with lattice QCD results (ASWZ24, LPC23) and lies between previous phenomenological extractions (ART23, ART25). It shows good agreement with the EEC (Energy-Energy Correlator) extraction at large $b$.
• TMD PDFs: The extracted $x$-dependent TMD PDFs are smooth and positive. The evolution from $Q=5$ GeV to $Q=100$ GeV shows the expected broadening in $k_T$ space.
• Data Comparison: Both methods describe the data well. The primary visual difference is that the Bayesian prediction bands are slightly wider, offering a more conservative estimate of uncertainties, particularly for normalized collider datasets with small experimental errors.

5. Significance

• Robust Uncertainty Quantification: The study highlights that while central values are stable, the structure of uncertainties depends on the inference method. Bayesian inference provides a transparent framework for incorporating priors and marginalizing nuisance parameters (like collinear PDFs) naturally.
• Future-Proofing: The developed framework is highly adaptable. It can easily incorporate heterogeneous data sources (e.g., Lattice QCD inputs) and is scalable for future high-precision experiments like the Electron-Ion Collider (EIC).
• AI in Physics: This work serves as a blueprint for integrating AI not just for data analysis, but for model discovery (finding the right physics ansatz) and computational acceleration (emulators), setting a new standard for precision QCD phenomenology.

In conclusion, the paper successfully demonstrates that AI-assisted Bayesian inference is a viable, powerful, and complementary approach to traditional replica methods for extracting TMD PDFs, offering deeper insights into the parameter space and more robust uncertainty quantification.