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Learning Transverse Momentum Distributions from Raw Scattering Events via Conditional Diffusion
Jitao Xu, Christopher Cocuzza, Kevin Braga, Daniel Lersch, Nobuo Sato, Yaohang Li
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Summary
This paper introduces a conditional diffusion model framework to extract Transverse Momentum Dependent Parton Distribution Functions (TMD PDFs) directly from raw Semi-Inclusive Deep Inelastic Scattering (SIDIS) event kinematics. By utilizing a PointNet-Pool encoder and a 2D UNet-based diffusion process, the method bypasses traditional parameterized functional forms and iterative fitting. Evaluated on simulated CLAS12 kinematics, the model successfully recovers TMDs with quantifiable uncertainties that scale appropriately with event statistics, demonstrating robustness even in low-statistics regimes relevant to future Electron-Ion Collider experiments.
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Relation Signals (6)
Conditional Diffusion Model → extracts → TMD PDF
confidence 95% · We present a conditional diffusion model that learns to map raw SIDIS event kinematics directly to TMD PDFs
SIDIS → providesdatafor → TMD PDF
confidence 95% · Extracting transverse momentum dependent parton distribution functions (TMD PDFs) from semi-inclusive deep inelastic scattering (SIDIS) data
CLAS12 → collectsdatafrom → SIDIS
confidence 90% · With the CLAS12 detector at Jefferson Lab already collecting high-statistics SIDIS data
Conditional Diffusion Model → uses → PointNet-Pool
confidence 90% · events, which are encoded using a permutation-invariant PointNet-Pool architecture
Conditional Diffusion Model → uses → 2D UNet
confidence 90% · guides a 2D UNet that iteratively denoises Gaussian noise into a predicted TMD image
Electron-Ion Collider → willcollectdatafrom → SIDIS
confidence 90% · and the Electron-Ion Collider on the horizon... the volume and kinematic reach of available measurements will grow
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2608.27077v1
- Canonical: https://arxiv.org/abs/2608.27077v1
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Learning Transverse Momentum Distributions from Raw Scattering Events via Conditional Diffusion Jitao Xu Christopher Cocuzza Kevin Braga Affiliation: Department of Computer Science, Old Dominion University Affiliation: Department of Physics, William & Mary Daniel Lersch Nobuo Sato Yaohang Li Affiliation: Department of Computer Science, Old Dominion University Affiliation: Thomas Jefferson National Accelerator Facility Abstract 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 Transverse momentum dependent parton distribution functions (TMD PDFs) describe the joint distribution of quarks and gluons in longitudinal momentum fraction x and transverse momentum T k_T inside the proton, providing a three-dimensional picture of nucleon structure in momentum space (Collins, 2011). Semi-inclusive deep inelastic scattering (SIDIS) is one of the primary channels for accessing TMDs experimentally. With the CLAS12 detector at Jefferson Lab already collecting high-statistics SIDIS data (Burkert and others, 2020) and the Electron-Ion Collider on the horizon (Abdul Khalek and others, 2022), the volume and kinematic reach of available measurements will grow substantially over the coming decade. Standard TMD extractions parameterize the non-perturbative component of the distribution, then fit the parameters to binned cross-section data by χ2χ^2 minimization (Collins, 2011). This strategy has produced reliable global fits, but the results depend on the chosen functional form, and propagating uncertainties through the full theory chain, where Sudakov evolution, matching corrections, and Fourier–Bessel transforms, remains expensive. Motivated by these limitations, we explore an alternative extraction strategy based on conditional diffusion models (Ho et al., 2020; Alghamdi et al., 2025). The idea is to learn the map from raw SIDIS event kinematics directly to the TMD, bypassing explicit parameterization. Specifically, the model represents the TMD on a discretized (x,bT)(x,b_T) grid (here bTb_T is the Fourier conjugate of T k_T; the two carry the same information) and learns to denoise it conditioned on a variable-size set of observed events, which are encoded using a permutation-invariant PointNet-Pool architecture (Qi et al., 2017) followed by a transformer encoder. Because the diffusion process is inherently stochastic, posterior uncertainty is obtained simply by drawing multiple samples, with no additional inference step required. We test the framework on simulated SIDIS data at CLAS12 kinematics. The extracted TMDs agree well with the ground truth, and the uncertainty bands narrow as the number of conditioning events increases, consistent with the expected statistical scaling. 2 Method Figure 1: Overview of the proposed framework. A variable-size set of raw SIDIS events is first mapped to per-point features by a shared PointNet convolution layer, then aggregated through six-way global pooling and refined by a lightweight Transformer encoder to produce a fixed-dimensional conditioning vector. This vector, together with a sinusoidal timestep embedding, guides a 2D UNet that iteratively denoises Gaussian noise into a predicted TMD image f(x,bT)f(x,b_T). Physics pipeline. The unpolarized SIDIS differential cross section at leading order is computed from the input TMD PDF f(x,bT)f(x,b_T) through a chain of physical operations (Collins, 2011; Collins et al., 1985): (i) perturbative Sudakov evolution from the initial scale to f(x,Q2,bT)f(x,Q^2,b_T), (i) convolution with the TMD fragmentation function D(z,Q2,bT)D(z,Q^2,b_T), taken from a prior extraction and held fixed throughout, and (i) Fourier–Bessel transform from bTb_T-space to transverse momentum qTq_T-space, yielding the leading-order structure function. Events (x,Q2,z,qT,ϕ)(x,Q^2,z,q_T,φ) are then sampled from the resulting cross section via an inverse transform sampler. To enable efficient evaluation and learning of the SIDIS cross section, we adopt a discretized representation of the underlying TMDs using a localized basis expansion. Specifically, functions such as f~(x,bT,Q2) f(x,b_T,Q^2) and D~(z,bT,Q2) D(z,b_T,Q^2) are expressed as linear combinations of basis functions defined on a grid in (x,Q2,bT)(x,Q^2,b_T) space. This representation allows all subsequent operations in the SIDIS pipeline—including evolution, convolution, and Fourier–Bessel transforms—to act linearly on the basis functions rather than directly on the unknown coefficients. As a result, computationally expensive operators can be precomputed and reused, reducing the full pipeline to a sequence of tensor contractions in coefficient space. This structure is particularly well-suited for generative modeling, where the diffusion model learns to produce physically consistent coefficient configurations that are subsequently mapped to observable distributions through the fixed physics pipeline. Importantly, because the diffusion model interfaces with the theory solely through simulated events, the pipeline is not required to be differentiable or written in the same programming language (Braga et al., 2025), which makes the framework readily applicable to other physics pipelines. Conditional diffusion model. The overall model architecture is illustrated in Figure 1. We employ a DDPM (Ho et al., 2020) with a 2D UNet backbone (Ronneberger et al., 2015) to model the distribution of TMD images. The UNet operates on (20×30)(20× 30) images with base channel width 96 and multipliers (1,2,4)(1,2,4), using a cosine noise schedule (Nichol and Dhariwal, 2021) with T=100T=100 diffusion steps. The input TMD is transformed via log(1+f) (1+f) to compress the dynamic range before diffusion. During training, a clean TMD image x0x_0 is corrupted by Gaussian noise according to xt=α¯tx0+1−α¯tϵ,ϵ∼(,),x_t= α_t\,x_0+ 1- α_t\, ε, ε (0,I), (1) where α¯t=∏s=1t(1−βs) α_t= _s=1^t(1- _s) is the cumulative noise schedule. The model ϵθ ε_θ is trained to predict the added noise by minimizing ℒ=x0,t,ϵ[‖ϵθ(xt,t,c)−ϵ‖2],L=E_x_0,\,t,\, ε [\| ε_θ(x_t,t,c)- ε\|^2 ], (2) where c denotes the conditioning embedding from the event encoder. At inference, new TMD samples are generated by iteratively denoising from xT∼(,)x_T (0,I). Event conditioning. To condition on a variable-size set of NevN_ev observed events, we use a PointNet-Pool encoder (Qi et al., 2017). Each 5D event is processed by shared 1D convolutions (5→64→128→1285→ 64→ 128→ 128), followed by six-way global pooling that produces a fixed 641-dimensional summary vector: hpool=[maxihi,meanihi,minihi,stdihi,stdihi/N,logN],h_pool= [ _ih_i,\;mean_ih_i,\; _ih_i,\;std_ih_i,\;std_ih_i/ N,\; N ], (3) where hih_i are the per-point features and N is the number of events. This vector is projected to 128 dimensions via a residual MLP. The event embedding is added to the sinusoidal timestep embedding and injected into each UNet residual block. 3 Experiments 3.1 Setup Figure 2: Evolved TMD f(x,Q2,bT)f(x,Q^2,b_T) on the full (x,bT)(x,b_T) grid at a representative Q2Q^2 slice. (a) Theory ground truth. (b–d) Diffusion model predictions (ensemble mean over 128 noise replicas) conditioned on Nev=1,000N_ev=1,000, 10,00010,000, and 100,000100,000 events, respectively. We evaluate the conditional diffusion model on SIDIS at Ebeam=11E_beam=11 GeV, matching CLAS12 kinematics at Jefferson Lab (Burkert and others, 2020). The TMD PDF is parameterized on a 20×3020× 30 grid in (x,bT)(x,b_T) with x∈[0.048,1.0]x∈[0.048,1.0] (log-spaced) and bT∈[0.001,6.0]b_T∈[0.001,6.0] GeV-1. The SIDIS theory module evolves the input TMD through perturbative Sudakov resummation, operator product expansion (OPE) corrections, and non-perturbative modeling, then computes differential cross section via Fourier transform from bTb_T-space to qTq_T-space. Events are sampled from the resulting cross section using an inverse transform sampler. Training uses a pre-generated parametric dataset of 50,000 TMD images with randomized non-perturbative parameters, optimized with Adam (Kingma and Ba, 2015) (learning rate 5×10−55× 10^-5, batch size 128) for 1,000 epochs. To study the dependence on the number of conditioning events, we train three models with Nev∈1,000, 10,000, 100,000N_ev∈\1,000,\ 10,000,\ 100,000\, all other hyperparameters held fixed. 3.2 Results Figure 2 shows the evolved TMD f(x,Q2,bT)f(x,Q^2,b_T) on the full (x,bT)(x,b_T) grid, comparing the theory ground truth to the model predictions at each conditioning event count. The model progressively recovers the two-dimensional structure of the distribution as NevN_ev increases. Figure 3: This figure shows one-dimensional slices of the two-dimensional TMD in Figure 2, at fixed x as a function of bTb_T. Evolved TMD slices at epoch 1,000 for three conditioning event counts. Dark gray: theory ground truth. Blue line and band: diffusion model mean ± 3σ±\,3σ over 128 noise replicas. Bottom panels: absolute model standard deviation σ. As NevN_ev increases, the prediction converges to the true TMD with narrowing uncertainty. Figure 3 shows evolved TMD slices f(x,Q2,bT)f(x,Q^2,b_T) vs. bTb_T at two representative x values, comparing the theory ground truth (dark gray) to the diffusion model prediction (blue) with ±3σ± 3σ model uncertainty bands from 128 independent diffusion samples. We summarize the spread of the diffusion posterior by ⟨σ⟩ σ , the standard deviation across 128 independent samples averaged over all (x,bT)(x,b_T) grid points. With Nev=1,000N_ev=1,000 events, the model captures the overall shape of the TMD but exhibits a broad uncertainty band (⟨σ⟩≈0.020 σ ≈ 0.020), reflecting the limited statistical information in a small event sample. At Nev=10,000N_ev=10,000, the mean closely tracks the theory curve across the full bTb_T range with uncertainty narrowing to ⟨σ⟩≈0.010 σ ≈ 0.010. At Nev=100,000N_ev=100,000, model and theory are nearly indistinguishable, with ⟨σ⟩≈0.004 σ ≈ 0.004. This scaling behavior is physically expected: the statistical precision of the conditioning events improves as ∼1/Nev 1/ N_ev, and the diffusion model faithfully translates this into correspondingly tighter posterior predictions. Crucially, even at Nev=1,000N_ev=1,000—an event count accessible in sparse kinematic bins or early-phase experiments—the model produces a reasonable TMD estimate with informative uncertainty. This demonstrates that the conditional diffusion framework can operate in data-scarce regimes relevant to upcoming EIC measurements (Abdul Khalek and others, 2022), where many (x,Q2)(x,Q^2) bins will have limited statistics. 4 Discussion and Conclusion We have demonstrated that a conditional DDPM can extract TMD PDFs from raw SIDIS event kinematics with informative uncertainties. The PointNet-Pool encoder provides an effective, permutation-invariant interface between variable-size event sets and the diffusion model, enabling natural uncertainty quantification through repeated stochastic sampling. The model’s performance scales gracefully with the number of conditioning events, producing informative posteriors even in low-statistics regimes. Several directions remain for future work: extending to next-to-leading order (NLO) theory, incorporating polarized observables, training on real experimental data from CLAS12, scaling to large-scale multi-GPU training, and applying the framework to projected Electron-Ion Collider pseudo-data with realistic detector effects. Acknowledgments and Disclosure of Funding This work is partially supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Computing (SciDAC) program, under contracts DE-AC02-06CH11357, DE-AC05-06OR23177, and DE-SC0023472. References Abdul Khalek et al. (2022) R. Abdul Khalek et al. Science requirements and detector concepts for the Electron-Ion Collider: EIC yellow report. Nuclear Physics A 1026, p. 122447. Cited by: §1, §3.2. Alghamdi et al. (2025) T. Alghamdi, J. Xu, N. Ramachandra, N. Sato, and Y. Li Towards an event-level analysis in hadronic physics using generative AI-based surrogates. In 2025 IEEE 37th International Conference on Tools with Artificial Intelligence (ICTAI), p. 445–452. External Links: Document Cited by: §1. Braga et al. (2025) K. Braga, M. Diefenthaler, S. Goldenberg, D. Lersch, Y. Li, J.-W. Qiu, K. Rajput, F. Ringer, N. 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