Impact Parameter Space Interpretation for Generalized Parton Distributions

TL;DR

The paper develops a rigorous interpretation of impact parameter dependent parton distributions as transverse spatial densities by relating them to the xi=0 Fourier transform of GPDs. It establishes a positive, probabilistic interpretation for $q(x,\mathbf{b}_\perp)$ and connects it to the helicity-conserving GPDs $H_q$ and $\tilde{H}_q$, while clarifying the role of the helicity-flip distribution $E_q$ in inducing transverse distortions and their connection to anomalous magnetic moments. It also discusses limitations from nonzero skewness, provides modeling constraints for realistic $H_q$, and offers an overlap-based LCWF framework to compute GPDs, with implications for phenomena such as transverse hyperon polarization. Together, these results illuminate how partons’ longitudinal momentum fractions and transverse positions encode the three-dimensional structure of the nucleon in the infinite momentum frame. The work supplies both theoretical bounds and practical modeling guidance to leverage GPDs for tomographic nucleon imaging.

Abstract

The Fourier transform of generalized parton distribution functions at xi=0 describes the distribution of partons in the transverse plane. The physical significance of these impact parameter dependent parton distribution functions is discussed. In particular, it is shown that they satisfy positivity constraints which justify their physical interpretation as a probability density. The generalized parton distribution H is related to the impact parameter distribution of unpolarized quarks for an unpolarized nucleon, H-tilde is related to the distribution of longitudinally polarized quarks in a longitudinally polarized nucleon, and $E$ is related to the distortion of the unpolarized quark distribution in the transverse plane when the nucleon has transverse polarization.

Figures (6)

• Figure 1: Impact parameter dependent parton distribution $u(x,{\bf b_\perp})$ for the simple model (\ref{['eq:model']}).
• Figure 2: Comparison of a non-rotating sphere that moves in the $z$ direction with a sphere that spins at the same time around the $z$ axis and a sphere that spins around the $x$ axis When the sphere spins around the $x$ axis, the rotation changes the distribution of momenta in the $z$ direction (adds/subtracts to velocity for $y>0$ and $y<0$ respectively). For the nucleon the resulting modification of the (unpolarized) momentum distribution is described by $E(x,0,t)$.
• Figure 3: $u$ quark distribution in the transverse plane for $x=0.1$, $0.3$, and $0.5$ (\ref{['eq:model']}). Left column: $u(x,{\bf b_\perp})$, i.e. the $u$ quark distribution for unpolarized protons; right column: $u_X(x,{\bf b_\perp})$, i.e. the unpolarized $u$ quark distribution for 'transversely polarized' protons $\left|X\right\rangle = \left|\uparrow\right\rangle + \left|\downarrow\right\rangle$. The distributions are normalized to the central (undistorted) value $u(x,{\bf 0_\perp})$.
• Figure 4: Same as Fig. \ref{['fig:panelu']}, but for $d$ quarks.
• Figure 5: $P+P(\bar{P})\longrightarrow Y+\bar{Y}$ where the incoming $P$ (from bottom) is deflected to the left during the reaction. The $s\bar{s}$ pair is assumed to be produced roughly in the overlap region, i.e. on the left 'side' of the $Y$.