Source-linked AI summary
Learning Transverse Momentum Distributions from Raw Scattering Events via Conditional Diffusion
Jitao Xu, Christopher Cocuzza, Kevin Braga, Daniel Lersch, Nobuo Sato, Yaohang Li
TL;DR
TMD extraction traditionally relies on parameterized fits whose uncertainty propagation can be expensive. This paper uses conditional diffusion to map raw SIDIS events directly to TMD PDFs, finding agreement with simulated ground truth and tighter uncertainties as event counts grow, including at 1,000 events.
Problem
Traditional TMD extractions depend on chosen functional forms, while propagating uncertainties through the full theory chain remains expensive.
Method
A conditional diffusion model encodes variable-size raw SIDIS event sets and guides a UNet that generates TMD images through iterative denoising.
Results
The extracted TMDs agree well with ground truth, and uncertainty bands narrow as conditioning-event statistics increase.
Takeaways & Limitations
The framework produces informative posterior estimates even in low-statistics regimes, including 1,000 conditioning events.
Takeaways & Limitations
Future work must extend the framework to NLO theory, polarized observables, real CLAS12 data, and realistic EIC detector effects.
Abstract
from arXiv · showhide
Extracting transverse momentum dependent parton distribution functions (TMD PDFs) from semi-inclusive deep inelastic scattering (SIDIS) data is a central goal of the nucleon structure program at Jefferson Lab and the future Electron-Ion Collider. Traditional extraction methods rely on parameterized functional forms and iterative fitting, which can limit the flexibility of the resulting distributions and make uncertainty quantification cumbersome. We present a conditional diffusion model that learns to map raw SIDIS event kinematics directly to TMD PDFs, bypassing explicit functional assumptions. Evaluated on simulated SIDIS data at CLAS12 kinematics, the model recovers the underlying TMD with informative uncertainties that narrow steadily with increasing event statistics, and produces reliable estimates even with as few as 1,000 conditioning events, a statistics-limited regime directly relevant to ongoing and planned experiments.
1 Introduction
TMD extractions traditionally fit parameterized forms to binned SIDIS cross sections, while this work explores conditional diffusion to map raw events directly to TMDs. On simulated CLAS12 data, the method recovers the TMD and narrows uncertainty as event statistics increase.
- Motivation: Standard TMD extractions depend on chosen functional forms and require expensive uncertainty propagation through the theory chain.The chain includes Sudakov evolution, matching corrections, and Fourier–Bessel transforms.
- Proposed approach: Conditional diffusion learns a direct map from raw SIDIS event kinematics to discretized TMDs, bypassing explicit parameterization.The TMD is represented on an (x, bT) grid and conditioned on variable-size event sets.
- Proposed approach: Posterior uncertainty is obtained by drawing multiple diffusion samples without an additional inference step.
- Evaluation: Simulated SIDIS tests at CLAS12 kinematics show good agreement between extracted TMDs and ground truth.Uncertainty bands narrow as the number of conditioning events increases.
2 Method
The method combines a physics pipeline with conditional diffusion: event sets are encoded into a fixed-dimensional conditioning representation, while a UNet learns to denoise TMD images. The event encoder handles variable-size inputs, and inference generates samples by iterative denoising.
- Physics pipeline: The SIDIS pipeline evolves the input TMD, convolves it with a fixed fragmentation function, and Fourier–Bessel transforms it to produce the leading-order structure function.Events are then sampled from the resulting cross section.
- Physics pipeline: A localized basis expansion makes evolution, convolution, and Fourier–Bessel transforms linear operations on basis functions.The representation is defined on a grid in (x, Q2, bT) space.
- Conditional diffusion: A DDPM with a 2D UNet models TMD images after log(1 + f) transformation compresses their dynamic range.The UNet operates on 20 × 30 images with 100 diffusion steps.
- Conditional diffusion: The model is trained to predict Gaussian noise added to clean TMD images, using an event-derived conditioning embedding.
- Event conditioning: PointNet-Pool processes each event with shared convolutions and six-way pooling to create a fixed-dimensional summary that conditions the UNet.The summary is projected through a residual MLP and combined with the timestep embedding.
3 Experiments
The conditional diffusion model is evaluated on simulated SIDIS data at CLAS12 kinematics, recovering TMD structure while reducing posterior uncertainty as conditioning statistics increase. Even 1,000 events yield a reasonable estimate with informative uncertainty.
- 3 Experiments: Figure 2 compares theory ground truth with diffusion predictions conditioned on Nev = 1,000, 10,000, and 100,000 events.Each prediction is an ensemble mean over 128 noise replicas.
- 3 Experiments: The model is tested at Ebeam = 11 GeV using a 20 × 30 TMD grid in (x, bT) under CLAS12 kinematics.The grid spans x ∈[0.048, 1.0] and bT ∈[0.001, 6.0] GeV−1.
- 3 Experiments: The model progressively recovers the two-dimensional TMD structure as Nev increases.The comparison is performed over the full (x, bT) grid at a representative Q2 slice.
- 3 Experiments: The prediction converges toward the true TMD with narrowing uncertainty as the number of conditioning events increases.Figure 3 shows fixed-x slices as functions of bT, with mean ± 3σ bands from 128 noise replicas.
- 3 Experiments: At Nev = 1,000, 10,000, and 100,000, average posterior spread decreases from ⟨σ⟩≈0.020 to ⟨σ⟩≈0.010 and ⟨σ⟩≈0.004.At 100,000 events, the model and theory are nearly indistinguishable.
- 3 Experiments: Statistical precision improves approximately as ∼1/√Nev, producing a reasonable TMD estimate with informative uncertainty even at Nev = 1,000.This event count is described as relevant to sparse kinematic bins and early-phase experiments.
4 Discussion and Conclusion
The study demonstrates conditional DDPM extraction of TMD PDFs from raw SIDIS event kinematics with informative uncertainties. Its performance scales with conditioning-event count, while future work must extend the framework beyond the current simulated and theory scope.
- 4 Discussion and Conclusion: A conditional DDPM extracts TMD PDFs from raw SIDIS event kinematics with informative uncertainties.The approach uses a PointNet-Pool encoder for variable-size event sets and repeated stochastic sampling for uncertainty quantification.
- 4 Discussion and Conclusion: Performance scales gracefully with conditioning-event count, producing informative posteriors in low-statistics regimes.The conclusion identifies this behavior as a central demonstrated capability.
- 4 Discussion and Conclusion: Future work includes NLO theory, polarized observables, real CLAS12 data, large-scale multi-GPU training, and realistic EIC detector effects.These directions define the framework’s current scope boundary.