Unraveling hadron structure with generalized parton distributions

TL;DR

This review presents generalized parton distributions (GPDs) as a unified framework that encodes hadron structure across form factors, parton densities, and distribution amplitudes, offering a phase-space perspective akin to Wigner functions. It systematizes the twist-two operator basis, derives the full set of GPDs for different hadron spins, and elaborates on their key properties like forward limits, polynomiality, and symmetry constraints. The article details evolution, gauge-invariant operator definitions, and the role of double distributions and impact-parameter space in connecting GPDs to experimental observables such as DVCS and hard exclusive meson production, while highlighting how GPDs enable three-dimensional imaging and access to orbital angular momentum via the Ji sum rule. Together, these developments establish GPDs as a central tool for mapping the spatial and spin structure of the nucleon and for understanding the interplay of partonic degrees of freedom in exclusive processes. The work also surveys practical modeling approaches, including DD-based parametrizations and positivity constraints, to guide phenomenology and future experiments aiming at a full tomographic picture of hadron structure.

Abstract

The generalized parton distributions, introduced nearly a decade ago, have emerged as a universal tool to describe hadrons in terms of quark and gluonic degrees of freedom. They combine the features of form factors, parton densities and distribution amplitudes--the functions used for a long time in studies of hadronic structure. Generalized parton distributions are analogous to the phase-space Wigner quasi-probability function of non-relativistic quantum mechanics which encodes full information on a quantum-mechanical system. We give an extensive review of main achievements in the development of this formalism. We discuss physical interpretation and basic properties of generalized parton distributions, their modeling and QCD evolution in the leading and next-to-leading orders. We describe how these functions enter a wide class of exclusive reactions, such as electro- and photo-production of photons, lepton pairs, or mesons. The theory of these processes requires and implies full control over diverse corrections and thus we outline the progress in handling higher-order and higher-twist effects. We catalogue corresponding results and present diverse techniques for their derivations. Subsequently, we address observables that are sensitive to different characteristics of the nucleon structure in terms of generalized parton distributions. The ultimate goal of the GPD approach is to provide a three-dimensional spatial picture of the nucleon, direct measurement of the quark orbital angular momentum, and various inter- and multi-parton correlations.

Figures (62)

• Figure 1: The Wigner function for lowest-lying quantum states of the harmonic oscillator, $n = 0$, $n = 2$ and higher level $n = 10$. The peak for $n = 0$ state is the most probable phase-space point of a particle at rest. The most probable orbit of the quantum oscillator is shown by the outermost circular orbit in phase space from the solution of classical equations of motion for the classical oscillator, see $n = 10$.
• Figure 2: Mach-Zender interferometric scheme for the measurement of the quantum-mechanical Wigner distribution of a light mode. BS1 and BS2 denote the low-reflection beam splitters. The quantum state is prepared using the neutral density filter ND and a mirror mounted on a piezoelectric translator PZT. The electro-optic modulators EOM1 and EOM2 control, respectively, the amplitude and the phase of the point at which the Wigner function is measured. The signal field is focused on a single photoncounting module SPCM.
• Figure 3: Localization of the nucleon with a wave packet.
• Figure 4: Breit frame for the $\gamma^\ast p \to p'$ process.
• Figure 5: Hadronic tensor of deep inelastic scattering cross section determining the imaginary part of the forward Compton scattering amplitude $\gamma^\ast (q) N (p) \to \gamma^\ast (q) N (p)$.