Learning Transverse Momentum Distributions from Raw Scattering Events via Conditional Diffusion

In this work we demonstrate that a single transformer-based generative model can capture Standard Model structure spanning five decades of invariant mass, from the sub-GeV regime to the TeV continuum, a range that no single Monte Carlo sample covers. To achieve this we design ShellFlow, a Riemannian conditional flow matching model that, given the recorded event composition, generates each particle on its on-shell manifold. Its only physics priors are the on-shell condition and the invariant-mass formula. The model is trained on $\sim 10^{9}$ real $pp$ collision events from the ATLAS Open Data 13~TeV release and told nothing else. From a single training run, the model learns to reproduce all of the following: intra-particle kinematics, the dilepton resonances ($J/ψ$, $Υ$, $Z$) at their PDG positions, the leptonic Weinberg angle, the $W$ and top-quark masses, and inter-particle correlations that enter no training objective. A substantial fraction of the Standard Model is thus learnable directly from recorded collision data.

Training-free conditional diffusion provides a flexible alternative to task-specific conditional model training, but existing samplers often allocate computation inefficiently: independent guided trajectories can vary widely in quality, and additional function evaluations along a single trajectory may not recover from poor early decisions. We propose Tempered Guided Diffusion (TGD), an annealed sequential Monte Carlo framework for training-free conditional sampling with diffusion priors. TGD targets tempered posterior distributions over the clean signal, using noisy diffusion states only as auxiliary variables for proposing reconstructions and propagating particles. Particles are reweighted by incremental likelihood ratios, resampled, and propagated across noise levels, concentrating computation on trajectories plausible under both the prior and observation. Under idealized exact-reconstruction assumptions, full TGD yields a consistent particle approximation to the posterior as the number of particles grows. For expensive reconstruction tasks, Accelerated TGD (A-TGD) retains early particle exploration but prunes to a single high-likelihood trajectory partway through sampling. Experiments on a controlled two-dimensional inverse problem and image inverse problems show improved posterior approximation and favorable wall-clock speed-quality tradeoffs over independent multi-trajectory baselines.

We establish finite-sample guarantees for conditional diffusion models learned from data and implemented with finitely many denoising steps. Our approach aligns the score-training objective with the sampling grid, allowing statistical estimation and numerical discretization errors to be analyzed together. Under a smooth conditional density model, we derive high-probability bounds on the generated distribution's error and quantify the required number of score evaluations. For every smoothness index $β>0$, a suitable grid yields averaged conditional Kullback--Leibler (KL) divergence $\widetilde O_p(n^{-2β/(d_x+d_y+2β)})$ relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error $\widetilde O_p(n^{-β/(d_x+d_y+2β)})$. Here $n$ is the sample size, $d_x,d_y$ are the data and conditioning dimensions, and errors are averaged over the conditioning variable. The total-variation bound matches the rate established in existing conditional distribution-estimation theory, up to logarithmic factors.