The Finite Moment Problem, Parton Distribution Functions, and Gaussian Processes

Recent algorithmic developments in Lattice Quantum Chromodynamics (LQCD) now enable the computation of parton distribution function (PDF) moments to unprecedented precision, providing first-principles insights into nuclear structure that are competitive with data-driven extractions. To address the challenge of reconstructing PDFs from these noisy moments, we propose a Bayesian framework revolving around Gaussian processes (GPs). We establish both the theoretical foundations and novel elements of this framework, and evaluate its performance across an assortment of phenomenological pion and nucleon PDF datasets, selected to capture the diverse range of behaviors spanned by valence, gluon, and sea quark PDFs. Our framework robustly reconstructs PDFs from a moderate number of moments, and resolves persistent issues in standard reconstruction approaches. We conclude by reconstructing the pion PDF using new LQCD moment data.