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Neural-Network extraction of TMDs with SIDIS data

This paper presents the first global analysis of unpolarized Transverse-Momentum-Dependent (TMD) distributions using a neural-network parametrization that simultaneously incorporates Drell-Yan and SIDIS data at N $^3$ LL accuracy, demonstrating that the inclusion of SIDIS data broadens the extracted TMDs while reducing uncertainties compared to Drell-Yan-only analyses and offering a model-independent framework for future high-precision experiments.

Original authors: Matteo Cerutti

Published 2026-06-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Matteo Cerutti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand the shape of a cloud. You can't just look at it from one angle; you need to see how the water droplets are moving and spread out in three dimensions. In the world of particle physics, scientists are trying to map out the "cloud" inside a proton (a building block of matter). Specifically, they want to know how the tiny particles inside (quarks) are moving not just forward, but also side-to-side. This side-to-side movement is called Transverse Momentum.

Here is a simple breakdown of what Matteo Cerutti's paper does:

1. The Problem: Rigid Maps

For a long time, scientists have tried to map these quark movements using mathematical formulas. Think of these formulas like a pre-drawn map. The problem is that these old maps were too rigid. They assumed the "cloud" of quarks had a specific, fixed shape. If the real cloud was slightly different, the map couldn't show it, leading to disagreements between different scientists' results.

2. The New Tool: A Flexible Neural Network

In this paper, the author introduces a new tool: a Neural Network (NN).

The Analogy: Instead of a rigid, pre-drawn map, imagine a clay sculpture. A neural network is like a sculptor who can mold the clay into any shape the data suggests, without forcing it into a pre-set box.
This flexibility allows the model to learn the true shape of the quark distribution directly from the data, rather than guessing the shape first.

3. The Experiment: Two Different Cameras

To build this map, the author used data from two different types of particle collisions, which act like two different cameras taking pictures of the same object:

Camera A (Drell-Yan): This is like smashing two heavy trucks (hadrons) together. It gives a clear picture of the quarks inside.
Camera B (SIDIS): This is like firing a bullet (an electron) at a truck and watching what pieces fly off. This gives a slightly different view that includes how the pieces break apart (fragmentation).

Previously, some studies only used Camera A. This study is special because it uses both cameras simultaneously to build a single, unified map.

4. The Results: A Wider, Clearer Picture

When the author combined the data from both cameras using the flexible clay-sculptor (Neural Network), they found two main things:

The Shape is Broader: When they added the "SIDIS" camera data, the resulting map of the quark cloud turned out to be wider than when they only used the "Drell-Yan" camera. It's as if the second camera revealed that the cloud spreads out more than previously thought.
The Uncertainty is Smaller: Because they had more data points from both cameras, they were more confident in their measurements. The "fuzziness" around the edges of their map got smaller compared to using just one camera. However, because the clay-sculptor is so flexible, the map is still a little bit "fuzzier" than the old, rigid pre-drawn maps. This is actually a good thing—it means the model is being honest about what it doesn't know, rather than pretending to be more precise than it is.

5. Why It Matters

The study proves that using these flexible, AI-like tools (Neural Networks) is a smart way to stop scientists from forcing their data into old, rigid boxes. It shows that by combining different types of experiments, we get a more accurate and honest picture of how matter is built.

The author notes that this approach will be very helpful for future, high-precision experiments at major facilities like Jefferson Lab and the Electron-Ion Collider, helping them get the most out of their data.

In short: The author used a flexible AI tool to combine two different types of particle collision data. This revealed that the internal structure of protons is wider than previously thought and provided a more honest, reliable map of how quarks move inside them.

Technical Summary: Neural-Network extraction of TMDs with SIDIS data

Problem Statement
Transverse-Momentum-Dependent (TMD) distributions offer a three-dimensional momentum-space map of hadron structure. While Drell-Yan (DY) processes constrain quark TMD parton distribution functions (PDFs), Semi-Inclusive Deep Inelastic Scattering (SIDIS) provides simultaneous access to TMD PDFs and fragmentation functions (TMD FFs). Despite advances in perturbative accuracy and global datasets, tensions persist between current extractions of unpolarized quark TMD PDFs from different global fits. These discrepancies are attributed partly to the limited flexibility of standard parametrizations used to model the nonperturbative sector of TMDs. Previous work demonstrated that Neural Networks (NNs) could reduce parameterization bias in DY-only analyses, but a global analysis incorporating SIDIS data was required to fully assess the constraints on TMD distributions.

Methodology
This work presents a global analysis of unpolarized TMD distributions utilizing a neural-network parametrization, extending a previous proof-of-concept study on DY data to include SIDIS measurements. The analysis operates at next-to-next-to-next-to-leading logarithmic (N3LL) accuracy.

Theoretical Framework: The differential cross sections for DY ( $h_A + h_B \to \ell^+ + \ell^- + X$ ) and SIDIS ( $\ell + h \to \ell' + h' + X$ ) are formulated in terms of TMD PDFs and FFs in impact-parameter space ( $b_T$ ). The TMD structure is factorized into perturbative coefficients (matching to collinear PDFs/FFs), evolution kernels (anomalous dimensions $\gamma$ and $K$ ), and a nonperturbative function ( $f_{NP}$ ).
Neural Network Parametrization: The nonperturbative component $f_{NP}$ $f_{N P}$ is split into an "intrinsic" part parametrized entirely by a neural network and an evolution term. The NN architecture is defined as [2, 10, 1], taking inputs of the longitudinal momentum fraction ( $x$ $x$ or $z$ $z$ ) and $b_T$ $b_{T}$ , with 10 hidden nodes and a sigmoid activation function.
- The total number of free parameters is 83: 41 for the TMD PDF nonperturbative function, 41 for the TMD FF nonperturbative function, and one parameter ( $g_2$ ) for the evolution.
- To prevent overfitting, the dataset is divided into training and validation samples of equal size.
Data and Fitting: The global dataset comprises 2,029 data points: 482 from DY measurements (Fermilab, RHIC, LHC) and 1,547 from SIDIS multiplicities (HERMES, COMPASS). Kinematic cuts are applied to satisfy TMD factorization requirements ( $q_T/Q < 0.2$ for DY).
Uncertainty Estimation: Experimental errors are propagated using the bootstrap method, fitting 100 Monte Carlo replicas of the experimental data. The best-fit parameters are obtained by minimizing a $\chi^2$ function.

Key Contributions and Results
The analysis successfully performs a simultaneous extraction of TMD PDFs and FFs, achieving a fit quality of $\chi^2/N_{dat} = 1.01$ . The primary findings include:

Impact of SIDIS Data on TMD Width: The inclusion of SIDIS data leads to broader unpolarized TMD PDFs compared to a DY-only NN extraction (NNDY). This broadening effect is consistent with observations in traditional parametrization models (e.g., MAP22).
Uncertainty Quantification: The uncertainties associated with the NN Global extraction are reduced relative to the DY-only case but remain larger than those obtained using traditional models (MAP22). The authors interpret this as evidence that the increased flexibility of the NN parametrization allows for a more robust and realistic assessment of uncertainties, avoiding the artificial tightening often seen in rigid functional forms.
Collins-Soper Kernel: The extracted Collins-Soper (CS) kernel (rapidity evolution anomalous dimension) from the NN Global analysis shows good agreement with the NNDY extraction and lattice QCD calculations. Notably, at large $b_T$ , the result is smaller than the NNDY extraction, which is consistent with the broader TMD PDFs observed in momentum space.
Data Consistency: The results indicate no significant tensions between DY and SIDIS data regarding the extraction of nonperturbative TMD evolution.

Significance
The paper demonstrates the potential of flexible neural-network parametrizations to reduce model dependence in global TMD analyses. By successfully incorporating SIDIS data alongside DY data at N3LL accuracy, the study provides a more comprehensive constraint on unpolarized TMD distributions. These results offer guidance for maximizing the physics potential of future high-precision measurements at Jefferson Lab (specifically the SoLID experiment) and the Electron-Ion Collider.