Parton Distribution Functions and their Generalizations

TL;DR

The paper surveys the progression from traditional parton distribution functions (PDFs) to their multi-dimensional generalizations—transverse momentum dependent distributions (TMDs), generalized parton distributions (GPDs), and generalized transverse momentum dependent distributions (GTMDs)—as a comprehensive framework for describing hadron structure in QCD. It lays out the theoretical foundations in the light-front formalism, operator product expansion, and QCD factorization, detailing how PDFs and their generalizations are defined, evolved, and related to observable processes such as DIS, SIDIS, DVCS, and Drell-Yan. It highlights how TMDs and GPDs reveal rich spin and spatial information, including angular momentum and mechanical properties through EMT form factors, and how GTMDs provide a unifying, phase-space view linked to Wigner distributions and orbital angular momentum. The article also surveys theoretical approaches (large-$N_c$ limits, models, lattice QCD) and outlines current experimental status and future prospects, underscoring the shift toward a three-dimensional tomographic understanding of the nucleon with significant implications for hadron structure and QCD phenomenology.

Abstract

This article is an introduction to parton distribution functions and their generalizations which describe the quark and gluon structure of hadrons, and can be measured in various high-energy scattering processes. We provide the theoretical background, highlight both historical and recent developments, explain the connections between the different functions, and expose in which processes these functions can be accessed and what we can learn from them about hadron structure.

Figures (22)

• Figure 1: A nucleon moving with the momentum $P_z \to \infty$ along the $z$-axis has a large light-front plus momentum $P^+=\frac{1}{\sqrt{2}}(P^0+P_z) \to \infty$. In this so-called infinite-momentum frame, the nucleon looks like a two-dimensional pancake due to extreme Lorentz contraction. Each of its constituents, called partons, carries a certain fraction $x$ of the nucleon's momentum, has some transverse momentum $\vec{k}_\perp$, and is located at a transverse position $\vec{b}_\perp$ relative to the center of the nucleon. The distributions of the partons in these variables, which further depend on the nucleon and parton polarizations, are described by parton distribution functions and their generalizations.
• Figure 2: Amplitudes of (a) elastic electron-nucleon scattering, $eN\to e'N'$, where electromagnetic form factors $F_i(t)$ are measured; (b) inclusive deep-inelastic scattering (DIS), $eN\to e'X$, from which PDFs are extracted; (c) semi-inclusive DIS, $eN\to e' h X$, where a specific hadron $h$ is detected and TMDs can be studied; (d) deeply virtual Compton scattering, $eN\to e'N'\gamma$, where GPDs are accessed.
• Figure 3: Graphical representation of the quark correlation function $\Phi^{q[\Gamma]}(P,k,\Delta)$ in Eq. (\ref{['GPCF']}) with the definitions of nucleon and quark momenta. For brevity we have omitted polarization labels for the initial-state and final-state nucleons.
• Figure 4: Experimentally observable functions (left panel), their interpretations (right panel) and inter-relations (indicated by the arrows). Note that the PDFs and the TMDs are densities in momentum space, indicated in the right panel as $\rho(x)$ and $\rho(x,\vec{k}_\perp)$, respectively.
• Figure 5: (a) The amplitude of deep-inelastic scattering (DIS) in the Bjorken-limit, where the electron scatters elastically off a charged parton carrying the momentum fraction $x \simeq x_B$ of the nucleon's momentum $P$. (b) The DIS cross section is proportional to the "handbag" diagram representing the product of the DIS amplitude (left of vertical line) and its complex conjugate (right of vertical line). The non-perturbative blob labeled PDF$(x)$ is the diagrammatical representation of a PDF and includes a summation over a complete set of states. (c) Amplitude for the Drell-Yan process, where a parton and antiparton from two colliding hadrons annihilate to produce a virtual photon. The photon momentum $q^\mu$ is reconstructed from its decay into a $\mu^+\mu^-$ pair in the limit $s=(P_1+P_2)^2\to\infty$ and $Q^2=q^2\to\infty$, with their ratio equal to $x_1 x_2 = Q^2/s$ and fixed. (d) The square of the Drell-Yan amplitude.