TMDs in the Lens of Generative AI: A Pixel-Based Approach to Partonic Imaging

This paper presents a novel, nonparametric pixel-based framework that leverages generative AI and Bayesian inference to simultaneously extract transverse momentum dependent (TMD) parton distributions and their evolution kernels, thereby enabling unbiased 3D partonic imaging while rigorously characterizing uncertainties and resolving inherent degeneracies.

The Gist

Imagine trying to figure out what a hidden object looks like just by looking at the shadow it casts on a wall. This is essentially what physicists are trying to do when they study Transverse Momentum Dependent (TMD) distributions. They want to create a 3D "map" of the tiny particles (quarks and gluons) inside a proton, but they can only see the "shadows" (data) created when these particles smash into each other at high speeds.

This paper introduces a new, smart way to solve this "shadow puzzle" using a mix of advanced mathematics and Generative AI. Here is a breakdown of their approach using simple analogies:

1. The Problem: The Blurry Shadow

In the past, scientists tried to guess the shape of the proton's internal map by assuming it looked like a specific, smooth curve (like a perfect bell shape). But the proton might not be that simple.

The paper argues that this is like trying to guess the shape of a complex sculpture just by looking at a blurry shadow. If you assume the shadow must come from a smooth ball, you might miss all the interesting bumps and dents. Furthermore, the math involved in turning the shadow back into the object is "ill-posed." This means that many different shapes could cast the exact same shadow. If you only have data from one specific angle (one energy level), there are parts of the object that are mathematically invisible to you, no matter how much data you collect. The authors call these invisible parts "Null TMDs"—features of the proton that the current data simply cannot "see."

2. The Solution: A Pixelated Approach

Instead of guessing a smooth curve, the authors decided to treat the proton's internal map like a digital image made of pixels.

• The Old Way: Trying to fit the whole image to a single formula (like saying "the whole picture is a circle").
• The New Way: Breaking the image down into a grid of 50 tiny squares (pixels). They let the data decide the brightness of each pixel individually. This is "nonparametric," meaning they don't force the data to fit a pre-made mold; they let the data speak for itself.

3. The Engine: Generative AI as a Detective

Because there are so many pixels (50) and the math is incredibly complex, checking every possible combination of pixel brightness would take longer than the age of the universe. To solve this, they used Generative AI (specifically a "Normalizing Flow").

Think of the AI as a super-smart detective who has seen millions of these shadow puzzles before.

1. Training: The AI learns the general rules of what a "reasonable" proton map looks like (it knows the physics constraints).
2. Sampling: Instead of guessing one answer, the AI generates thousands of possible "pixel maps" that could explain the shadow.
3. Filtering: It uses a statistical method (Metropolis-Hastings) to keep only the maps that match the experimental data perfectly and discard the ones that don't.

This allows them to not just find one best map, but to understand the uncertainty of the map. They can say, "We are 95% sure the pixel here is bright, but we are totally unsure about the pixel there."

4. The "Precision Floor" and the Multi-Scale Trick

The authors discovered a hard limit. Even with perfect data, if you only look at the shadow from one angle (one energy level), there is a "precision floor." You can't see the tiny details in the center of the proton because the math of the shadow (the Bessel transform) acts like a diffraction-limited lens. It filters out the high-frequency details.

The Breakthrough:
To see the hidden details, you need to look at the shadow from multiple angles (different energy levels).

• Analogy: Imagine trying to see the texture of a rough stone. If you shine a light from one side, you see some shadows. If you move the light around (change the energy), the shadows shift, revealing different textures.
• By combining data from four different energy levels, the AI can "triangulate" the proton's structure. The high-energy data provides the "high-frequency" information needed to resolve the tiny, central details that low-energy data misses.

5. The Complex Case: The Convolution

The paper also tested this on a more difficult scenario: the Structure Function.

• Analogy: Imagine the shadow isn't just the proton, but the proton plus a piece of glass (the fragmentation function) that distorts the image before it hits the wall.
• The authors showed their AI could successfully "deconvolve" (undo) the distortion caused by the glass and still reconstruct the original proton map, even though the glass was hiding some of the details.

Summary of Findings

• Null TMDs exist: There are parts of the proton's structure that are mathematically invisible to single-energy experiments. They remain "unconstrained" and are only defined by our theoretical assumptions, not by the data.
• Multi-scale is key: You cannot overcome this invisibility just by collecting more data at the same energy. You must collect data at different energies to "break the degeneracy" and see the full picture.
• AI works: This pixel-based, AI-driven method successfully reconstructed the proton's internal map in their tests, providing a much more honest and detailed picture of what we know (and what we don't know) about the proton's 3D structure.

In short, the authors built a new, flexible camera (the pixel-AI framework) and proved that to get a sharp, 3D photo of the proton's heart, you have to take pictures from many different distances, not just one.

Technical Summary

Technical Summary: TMDs in the Lens of Generative AI

Problem Statement
The extraction of Transverse Momentum Dependent (TMD) parton distribution functions (PDFs) from experimental data constitutes a classic inverse problem. Within the Collins-Soper-Sterman (CSS) formalism, TMDs are related to observable cross-sections via a Bessel (Hankel) transform, mathematically formulated as a Fredholm integral equation of the first kind. This problem is intrinsically ill-posed: experimental data are discrete, finite in range, and subject to statistical uncertainties. Consequently, infinitely many configurations in impact parameter ($b_T$) space can reproduce the same observables within error margins.

Current state-of-the-art global analyses typically rely on rigid analytical functional forms or neural networks with explicit regularization to constrain these solutions. However, these approaches often fail to formally address the existence of "null TMDs"—functional modes residing in the effective null space of the integral transform. These modes remain fundamentally invisible to physical observables due to the finite momentum reach of data, yet they are often implicitly constrained by the chosen functional form or regularization, potentially leading to an underestimation of true uncertainties in 3D partonic imaging.

Methodological Framework
This work introduces a nonparametric, pixel-based framework for Bayesian inference and imaging of TMDs, validated through a hybrid Normalizing Flow-driven Metropolis-Hastings (NF-MH) approach.

1. Pixel-Based Discretization: Instead of assuming a global functional form, the TMD distribution in $b_T$ space is discretized into a grid of nodal values ("pixels"). The continuous distribution is reconstructed using local interpolators (Lagrange polynomials) defined on these nodes. This allows the data to dictate the shape of the distribution with minimal a priori bias.
2. Bayesian Inference: The reconstruction is framed as a probabilistic inference problem. The posterior probability distribution $P(\tilde{f}|f)$ is determined using Bayes' theorem, combining a likelihood function (based on $\chi^2$ residuals between data and theory) with a prior distribution. The prior employs Tikhonov-style regularization (up to second order) to penalize unphysical oscillations and ensure smoothness, without enforcing a specific global shape.
3. Generative AI Sampling: To efficiently sample the high-dimensional posterior (50+ dimensions), the authors utilize a Normalizing Flow (NF) trained to approximate the target posterior density. The NF serves as a global proposal distribution for a Metropolis-Hastings (MH) algorithm. This hybrid strategy bypasses the slow exploration typical of local random walks in high dimensions, enabling exact sampling of the posterior while maintaining computational efficiency via GPU acceleration and auto-differentiation.
4. Singular Value Decomposition (SVD): The framework employs SVD of the kernel matrix to rigorously characterize the resolution limits. This decomposes the solution space into an "observable space" (constrained by data) and a "null space" (unconstrained by data, determined solely by the prior).

Key Contributions and Results
The methodology is validated through three closure tests of increasing complexity: a baseline Gaussian model, the $u$-quark TMD PDF, and the unpolarized $F_{UU,T}$ structure function (involving convolution with fragmentation functions).

• Identification of Null TMDs: The SVD analysis formally defines and quantifies "null TMDs." The study reveals that for single-scale measurements, the Bessel kernel effectively filters out high-frequency components corresponding to small $b_T$ (short distances). These components reside in the null space, meaning their reconstruction is driven entirely by the theoretical prior rather than experimental data. This creates an irreducible "precision floor" where increasing statistical precision ($N_{ev}$) does not narrow uncertainty bands in the small-$b_T$ region.
• Multi-Scale Resolution: The paper demonstrates that this resolution barrier is broken by multi-scale analysis. By simultaneously fitting data across multiple energy scales ($\mu^2$), the evolution of the TMDs provides access to different regions of the $k_T$ spectrum. High-energy data probe the high-momentum tails, which correspond to the short-distance structure in $b_T$ space. This "kinematic lever arm" effectively populates the previously inaccessible observable subspace, significantly reducing the null-space dominance and lowering the precision floor.
• Deconvolution of Convoluted Observables: In the $F_{UU,T}$ case, the framework successfully deconvolves the TMD PDF from the fragmentation function (FF). The analysis shows that while the FF modulates sensitivity, it can also act as a limiting aperture, masking large-$b_T$ structures. The multi-scale approach remains effective in recovering the intrinsic hadronic structure even in this convoluted scenario.
• Uncertainty Quantification: The NF-MH framework provides a complete probability density for the distribution rather than a single "best-fit" curve. Ensemble analyses confirm that the framework correctly captures statistical fluctuations and distinguishes between internal model precision and external statistical stability.

Significance and Claims
The authors claim this work represents the first integration of pixel-based discretization, generative AI, and SVD within a Bayesian context to solve the TMD inverse problem. The primary significance lies in the shift from point-wise optimization to a full probabilistic mapping of the solution space, which:

1. Removes Inherent Degeneracies: By explicitly identifying and quantifying null TMDs, the framework avoids the false confidence that can arise from rigid functional assumptions.
2. Enables Unbiased Imaging: The nonparametric approach allows for the extraction of the nonperturbative evolution kernel and intrinsic hadronic structure simultaneously, without imposing rigid shapes on the nonperturbative components.
3. Defines Resolution Limits: The study formally establishes that the resolution of the 3D partonic map is not merely a function of statistical precision but is fundamentally limited by the spectral coverage of the data in momentum space. It argues that achieving high-fidelity imaging of the nucleon core requires multi-scale data to overcome the diffraction limit imposed by the Bessel transform.

The paper concludes that this synergy between machine learning and multi-scale data establishes a new path toward high-fidelity 3D partonic imaging, providing a robust foundation for future global analyses at facilities like Jefferson Lab and the Electron-Ion Collider (EIC).