GPDs

Generalized parton distributions (GPDs) are functions in quantum chromodynamics that describe the combined momentum and spatial structure of partons inside hadrons. They extend ordinary parton distribution functions (PDFs) by encoding information about the correlation between parton momentum and transverse position, and they reduce to PDFs or elastic form factors in appropriate limits. GPDs are defined through off-forward (nonzero momentum transfer) matrix elements of light-cone quark and gluon operators.

GPDs depend on several kinematic variables: x, the average longitudinal momentum fraction carried by the parton; xi, the skewness parameter related to the longitudinal momentum transfer; t, the squared momentum transfer to the hadron; and Q^2 (or mu), the renormalization scale. In the forward limit where xi and t vanish, certain GPDs reduce to the usual PDFs, while the first x moments of GPDs yield elastic form factors. The Fourier transform of GPDs with respect to t provides information about parton distributions in the transverse plane, offering a three-dimensional view of the hadron.

The quark sector features GPDs H_q and E_q, and the axial counterparts tilde H_q and tilde E_q; gluon GPDs H_g, E_g (and their axial analogs) are also defined. In addition to these, chiral-odd GPDs such as H_T exist. GPDs obey theoretical constraints, including polynomiality and positivity bounds, and their x-m moments relate to form factors and spin observables. The impact-parameter representation gives a probabilistic interpretation at fixed x and Q^2.

Hard exclusive processes provide access to GPDs through factorization theorems. Deeply virtual Compton scattering (DVCS) and hard exclusive meson production couple a short-distance kernel to GPDs. GPDs evolve with scale via generalized DGLAP/ERBL equations.

Experimentally, GPDs are probed at facilities such as Jefferson Lab, HERA, and COMPASS, with lattice QCD computing moments. Future endeavors, notably the Electron-Ion Collider, aim to map GPDs comprehensively, enabling detailed 3D imaging of nucleon structure.