New Map of the Proton’s Inner Workings Unveiled with Updated GPD Parametrization

Deep inside the proton, quarks and gluons swirl in a three-dimensional dance that physicists have spent decades trying to chart. Now a team of theorists at the University of Virginia and Virginia Tech has released a significantly upgraded tool for that quest: a flexible, physics-constrained parametrization of generalized parton distributions, or GPDs, dubbed UVA2. Published in The European Physical Journal C, the work assembles constraints from high-precision deep inelastic scattering data, flavor-separated electromagnetic form factors, and cutting-edge lattice QCD calculations into a single coherent framework that describes how the proton’s constituents are distributed in both momentum and position space.

GPDs are the mathematical objects that go beyond the familiar parton distribution functions measured in inclusive deep inelastic scattering. While ordinary PDFs tell you the probability of finding a quark carrying a given fraction of the proton’s momentum, GPDs also encode what happens when the proton recoils, absorbing a momentum transfer that shifts it sideways. That extra information, accessed through processes like deeply virtual Compton scattering, where an electron knocks a virtual photon out of the proton and a real photon emerges, is what allows physicists to reconstruct genuinely three-dimensional portraits of nucleon structure, including the elusive orbital angular momentum carried by quarks and gluons.

The central difficulty is that experiments do not measure GPDs directly. What they measure are Compton form factors, convolutions of the GPDs with perturbatively calculable kernels that are sharply singular at particular points. Recovering the underlying GPDs from these convolutions is an inverse problem that is, in principle, intractable without additional input. The Virginia team’s strategy is to narrow the space of possible solutions by building the parametrization directly on the symmetries of quantum chromodynamics: crossing symmetry under the exchange of quark and antiquark momenta, polynomiality of the Mellin moments, and the area rules dictated by baryon number and momentum conservation.

The mathematical backbone of UVA2 is the reggeized spectator model, in which the proton is pictured as a struck parton bound to a remnant whose invariant mass is allowed to fluctuate. This yields compact analytic expressions for the two leading vector GPDs, H and E, for each quark flavor, including the valence u and d quarks, the antiquark sea, and the gluons. In a first for this framework, the authors provide fully analytic closed forms in all kinematic variables, eliminating the need for numerically expensive transverse momentum integrations and making the parametrization immediately usable in regression analyses, neural network training, and Monte Carlo event generators.

Constraining such a multidimensional function requires an enormous amount of input. In the forward limit, where the GPD H reduces to ordinary PDFs, the team fitted their parameters to reproduce the NNPDF21 parton distributions, checking the quality of the match with Kolmogorov-Smirnov tests. The t-dependence, describing how the distributions fall off with transverse momentum transfer, was then pinned down by fitting the flavor-separated Dirac and Pauli form factors extracted from lepton-nucleon scattering, together with the second Mellin moments recently computed in lattice QCD. The gluon sector was anchored to lattice determinations of the gluon form factors evaluated at a pion mass closer to the physical value than in previous versions.

All of these ingredients are defined at a common initial evolution scale of 0.58 square gigaelectronvolts, an improvement over earlier versions in which quarks and gluons started from inconsistent scales. From there, the distributions are evolved to higher energy scales using the leading-order off-forward DGLAP and ERBL evolution equations, which the authors solved with the Adams predictor-corrector method rather than the conventional Runge-Kutta approach. Because the right-hand side of the evolution equations involves convolution integrals that are notoriously expensive to evaluate, this choice of integrator delivers a substantial gain in computational speed, particularly in the low-momentum-fraction region that will dominate at future colliders.

The practical payoff comes in the form of predictions for the Compton form factors across a vast kinematic range, from the fixed-target regime of Jefferson Lab, where valence quarks dominate, all the way to the low Bjorken-x frontier of the upcoming Electron-Ion Collider, where sea quarks and gluons take over. The team’s calculations show that at EIC kinematics the Compton form factors are expected to be several orders of magnitude larger than at fixed-target facilities, and that the valence contribution is strongly suppressed in the collider setting. Crucially, because the flavor-singlet quark combination couples to gluons in the evolution equations, the gluon GPDs can in principle be extracted directly from DVCS data at the EIC, turning a machine designed to map gluon structure into a precision instrument for proton tomography.

The authors are careful about uncertainties. The error budget on their GPDs originates entirely from the fits to the t-dependence of the electromagnetic form factors, the lattice gluon form factors, and the second moments of the quark distributions, while the forward-limit parameters, obtained by matching to PDF parametrizations rather than raw data, carry no statistical errors and are instead validated through distribution-to-distribution comparisons. They flag this Hessian-based treatment as a first step, with maximum-likelihood and Markov-chain Monte Carlo analyses, along with neural network approaches, planned to properly capture correlations among the seven to eight parameters that define each flavor sector.

Everything is openly available. The UVA2 parametrization, including analytic forms at the initial scale and precomputed grids spanning momentum fractions from ten thousandths up to nearly one, momentum transfers up to four square gigaelectronvolts, and evolution scales up to a thousand square gigaelectronvolts, can be downloaded from a public GitHub repository in easily readable CSV formats. This openness matters, because GPD parametrizations are becoming shared infrastructure: they feed event generators, guide experimental analyses of exclusive processes, and provide the benchmark curves against which lattice QCD results and future DVCS measurements will be judged.

As the Electron-Ion Collider moves from blueprint to beamline, tools like UVA2 will determine how much physics can be squeezed out of the first data. By fusing elastic scattering, inclusive deep inelastic data, and lattice QCD into a single symmetry-respecting framework, the Virginia collaboration has delivered what amounts to a flexible, evolving atlas of the proton’s interior, one that promises to turn the coming decade of scattering experiments into a genuine three-dimensional census of the most abundant visible matter in the universe.

Subject of Research: Global parametrization of generalized parton distributions describing the three-dimensional quark and gluon structure of the proton

Article Title: Updated flexible global parametrization of generalized parton distributions from elastic and deep inelastic inclusive scattering data

Article References: Panjsheeri, Z., Adams, D. Q., Khawaja, A., Pandey, S., Tezgin, K., & Liuti, S. (2026). Updated flexible global parametrization of generalized parton distributions from elastic and deep inelastic inclusive scattering data. The European Physical Journal C, 86(9), Article 1078. https://doi.org/10.1140/epjc/s10052-026-15872-0

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15872-0

Keywords: generalized parton distributions, proton structure, quantum chromodynamics, deeply virtual Compton scattering, lattice QCD, parton distribution functions, Electron-Ion Collider, QCD evolution, elastic form factors, Compton form factors, nucleon tomography, reggeized spectator model