Problem

Transverse-momentum-dependent (TMD) parton distribution functions (PDFs) are essential for understanding the three-dimensional momentum structure of protons, particularly in multi-scale processes like Drell-Yan production. A significant challenge in this field is the accurate extraction of TMD PDFs and the reliable quantification of their associated uncertainties. While traditional methods, such as the Monte Carlo replica method, are widely adopted, Bayesian inference offers a valuable alternative for uncertainty quantification. This work aims to conduct a comprehensive extraction of unpolarized quark TMD PDFs from Drell-Yan data using a Bayesian inference framework enhanced by artificial intelligence, and to directly compare these results with those from a replica-based analysis.

Method

The analysis employs a sophisticated Bayesian inference framework, augmented by artificial intelligence (AI) throughout the process. The theoretical calculations are performed at next-to-next-to-next-to-leading order (N3LO) in perturbative Quantum Chromodynamics (QCD), combined with next-to-next-to-next-to-next-to-leading logarithmic (N4LL) resummation accuracy. An AI-driven iterative procedure systematically explores and ranks potential functional forms for the nonperturbative components of TMD PDFs and the Collins-Soper evolution kernel, guided by chi-squared fits and fundamental physics constraints. To overcome the computational intensity of repeated theory evaluations during Bayesian inference, a machine-learning emulator is developed as a surrogate model for TMD cross-sections, enabling efficient posterior sampling through an affine-invariant Markov Chain Monte Carlo (MCMC) ensemble. For direct comparison, a replica-based analysis is also conducted within the identical theoretical and phenomenological setup. The nonperturbative modeling, including the Sudakov factor and Collins-Soper kernel, utilizes specific functional forms identified through the AI-driven exploration. Experimental Drell-Yan data from fixed-target, RHIC, and LHC experiments are incorporated, with careful kinematic selections to ensure adherence to the low-transverse momentum regime. The Bayesian approach employs an independent Beta prior for each parameter, and a trust score based on the emulator's ensemble spread is used to ensure reliability, falling back to exact theory calculations when predictive uncertainty is high.

Results

Both the Bayesian and replica inference methods yielded consistent central fits, achieving an overall fit quality of approximately χ²/N ≈ 1. Although the mean parameter values from the two approaches settled into different local minima, their resulting fit qualities remained comparable (χ²/N = 1.02 for the replica fit and 1.03 for the Bayesian fit). The extracted Collins-Soper kernels from both methods were found to be compatible, exhibiting overlapping 68% uncertainty bands across the entire momentum space and aligning well with existing phenomenological and lattice QCD determinations. The extracted transverse-momentum-dependent PDFs in both position and momentum space were smooth and physically consistent, showing the expected suppression at large transverse distance, which became more pronounced at higher hard scales, and displaying evolution-driven broadening in transverse momentum. When comparing theoretical predictions against experimental data, the central curves from both analyses were highly consistent, with differences primarily observed in the widths of the uncertainty bands. Notably, the Bayesian uncertainty bands were, on average, 1.23 times wider than those from the replica method, suggesting a slightly more conservative uncertainty estimate. Furthermore, the Gaussian/Hessian approximation for marginal uncertainties proved to be more accurate for the Bayesian fit, indicating its potential suitability for integration into other Gaussian-statistical fitting pipelines.

Implications

This research demonstrates that Bayesian inference offers a valuable and complementary approach to uncertainty quantification in TMD extractions, providing a transparent framework for exploring the complex parameter space and integrating diverse data sources. The successful integration of AI-assisted tools, including an AI-driven procedure for exploring functional forms and a machine-learning emulator for cross-section calculations, significantly streamlines and enhances the efficiency of the analysis workflow, enabling more extensive and systematic explorations than traditional manual methods. The framework developed in this study can be further extended to incorporate heterogeneous inputs, such as constraints from lattice QCD and future high-precision measurements from facilities like the Electron-Ion Collider. This combination of AI-assisted modeling with probabilistic inference opens up new avenues for developing more flexible and less biased parametrizations of TMDs, ultimately contributing to more precise studies of nucleon structure and unlocking the full scientific potential of upcoming experimental programs.