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Generalized parton distributions in impact parameter space

M. Diehl

TL;DR

Diehl analyzes generalized parton distributions in impact-parameter space, including nonzero skewness ξ, to reveal how fast-moving hadrons encode both longitudinal and transverse structure. By developing an overlap representation with light-cone wave functions, he derives ξ-dependent, impact-parameter–space GPDs and shows how Lorentz invariance enforces a transverse shift of the proton center of momentum for ξ≠0, linking these distributions to elastic form factors and positivity constraints. The work contrasts GPDs with k_T-dependent distributions, clarifying how hard processes probe correlated parton positions and momenta in a two-scale picture defined by μ (transverse resolution) and t (transverse localization). This framework provides a spatial imaging tool for nucleon structure and illuminates how parton distributions encode multidimensional information beyond the one-dimensional densities.

Abstract

We study generalized parton distributions in the impact parameter representation, including the case of nonzero skewness xi. Using Lorentz invariance, and expressing parton distributions in terms of impact parameter dependent wave functions, we investigate in which way they simultaneously describe longitudinal and transverse structure of a fast moving hadron. We compare this information with the one in elastic form factors, in ordinary and in k_T dependent parton distributions.

Generalized parton distributions in impact parameter space

TL;DR

Diehl analyzes generalized parton distributions in impact-parameter space, including nonzero skewness ξ, to reveal how fast-moving hadrons encode both longitudinal and transverse structure. By developing an overlap representation with light-cone wave functions, he derives ξ-dependent, impact-parameter–space GPDs and shows how Lorentz invariance enforces a transverse shift of the proton center of momentum for ξ≠0, linking these distributions to elastic form factors and positivity constraints. The work contrasts GPDs with k_T-dependent distributions, clarifying how hard processes probe correlated parton positions and momenta in a two-scale picture defined by μ (transverse resolution) and t (transverse localization). This framework provides a spatial imaging tool for nucleon structure and illuminates how parton distributions encode multidimensional information beyond the one-dimensional densities.

Abstract

We study generalized parton distributions in the impact parameter representation, including the case of nonzero skewness xi. Using Lorentz invariance, and expressing parton distributions in terms of impact parameter dependent wave functions, we investigate in which way they simultaneously describe longitudinal and transverse structure of a fast moving hadron. We compare this information with the one in elastic form factors, in ordinary and in k_T dependent parton distributions.

Paper Structure

This paper contains 15 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lorentz invariance and its consequence
  3. From momentum transfer to impact parameter
  4. Probing transversely localized protons
  5. Discussion
  6. The overlap representation in ${\mathbf{b}}$-space
  7. Fock space and wave functions
  8. The wave function representation of GPDs
  9. Positivity constraints
  10. Special cases
  11. The limit $\xi=0$
  12. Form factors
  13. A tale of two scales
  14. Unintegrated parton distributions
  15. Conclusions

Figures (2)

  • Figure 1: Representation of a GPD in impact parameter space. Plus-momentum fractions refer to the average proton momentum $\frac{1}{2}(p+p')$ and are indicated above or below lines. The region $\xi\le x\le 1$ is shown in (a), and the region $|x|\le\xi$ in (b).
  • Figure 2: Impact parameter representation of an unintegrated parton distribution. ${\mathbf{z}}$ is the Fourier conjugate variable to the transverse momentum ${\mathbf{k}}$ of the struck parton.