TMDs in the Lens of Generative AI: A Pixel-Based Approach to Partonic Imaging
This work introduces a novel, nonparametric pixel-based framework for the Bayesian inference and imaging of transverse momentum dependent (TMD) parton distributions. The methodology is built upon a fully differentiable framework that integrates TMD evolution with the Collins-Soper-Sterman formalism, enabling the simultaneous extraction of partonic distributions and the nonperturbative evolution kernel. To achieve efficient and exact sampling of the high-dimensional posterior, we leverage generative AI through a hybrid normalizing flow-driven Metropolis-Hastings approach. The framework is validated through multi-scale closure tests of increasing complexity, ranging from basic functional models to convoluted structure functions. Using singular value decomposition (SVD), we rigorously characterize the uncertainty of the reconstructed distributions and reveal the existence of null TMDs, which are functional components in the null space of the kernel that remain unconstrained by observables. The new framework provides the first integration of pixel-based discretization, generative AI, and SVD within a Bayesian context to solve the TMD inverse problem. This synergy between machine learning and multi-scale data removes inherent degeneracies and enables unbiased 3D partonic imaging.