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Rapidity-Dependent Spin Decomposition of the Nucleon

Florian Hechenberger, Kiminad A. Mamo, Ismail Zahed

Abstract

We show that the two-dimensional Fourier transform of the generalized parton distributions (GPDs) has two distinct interpretations: at zero skewness ($η=0$) it yields the familiar impact-parameter density, while at finite skewness ($η\neq 0$) it encodes a genuine parton-nucleon correlation whose norm decreases predictably with the rapidity gap $Δy = 2\mathrm{artanh} η$. This rapidity dependence produces universal rapidity-modified Ji identities linking helicity, orbital, and total angular momenta analytically. Using linear open- and closed string Regge trajectories constrained by empirical PDFs, spectroscopy and form factor data, we obtain the leading twist GPDs $H,E,\widetilde{H}$ across the full $(x,η,t)$ range. Numerical Mellin Barnes inversion agrees with existing lattice data and yields rapidity resolved predictions for Jefferson Lab 12 GeV, the Electron Ion Collider, and forthcoming lattice studies.

Rapidity-Dependent Spin Decomposition of the Nucleon

Abstract

We show that the two-dimensional Fourier transform of the generalized parton distributions (GPDs) has two distinct interpretations: at zero skewness () it yields the familiar impact-parameter density, while at finite skewness () it encodes a genuine parton-nucleon correlation whose norm decreases predictably with the rapidity gap . This rapidity dependence produces universal rapidity-modified Ji identities linking helicity, orbital, and total angular momenta analytically. Using linear open- and closed string Regge trajectories constrained by empirical PDFs, spectroscopy and form factor data, we obtain the leading twist GPDs across the full range. Numerical Mellin Barnes inversion agrees with existing lattice data and yields rapidity resolved predictions for Jefferson Lab 12 GeV, the Electron Ion Collider, and forthcoming lattice studies.

Paper Structure

This paper contains 17 sections, 25 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conventions and basic definitions
  3. Conformal moments and string-based parametrization
  4. Mellin--Barnes representation
  5. Forward limit and Reggeized Mellin transform
  6. Finite skewness and analytic continuation
  7. Evolution and MB inversion
  8. Spatial tomography at arbitrary skewness
  9. Impact-parameter densities and their interpretation
  10. Norm of the correlation and rapidity dependence
  11. Spin tomography and rapidity-modified Ji identities
  12. Ji decomposition from conformal moments
  13. Spin budget and transverse densities at $\eta=0$
  14. Finite skewness: rapidity-modified Ji identities
  15. Conclusions
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Non-singlet/isovector helicity (a) and spin-flip (b) GPDs reconstructed from the MB representation \ref{['eq:MB']} using the conformal moments defined in Eqs. \ref{['eq:ReggeMellin']}--\ref{['eq:SkewnessRestore2']}, compared with lattice data Alexandrou:2020zbeHolligan:2023jqh (filled circles). Theory bands are at $\mu=2\,$GeV after NLO evolution. Vertical dashed lines mark $x=\pm\eta$ (ERBL--DGLAP boundaries).
  • Figure 2: Spatial tomography at $\mu=2$ GeV for non-singlet transverse densities $\rho^{u/d}_{n}(\bm b_\perp;\eta)$ (with $n=1,2$) defined in Eq. \ref{['eq:bspaceFT']} for a nucleon polarized along $+\hat{\bm y}$. Panel (a) shows $\eta=0$ (impact-parameter densities). Panel (b) shows $\eta=0.33$, corresponding to a rapidity gap $\Delta y=2\,\operatorname{artanh}(\eta)\simeq 0.69$; the overall norms decrease according to Eq. \ref{['eq:NormVariationFinal']}.
  • Figure 3: Impact-parameter distributions for gluon (a), sea- (b), up- (c), and down- (d) quark helicity densities at $\mu=2$ GeV and $\eta=0$ obtained from Eq. \ref{['eq:Ji_decomp']}.
  • Figure 4: Impact-parameter distributions for up- (a) and down-quark (b) orbital angular momentum and up- (c) and down-quark (d) spin--orbit correlations at $\mu=2$ GeV and $\eta=0$. The densities follow from Eqs. \ref{['eq:Ji_decomp']} and \ref{['eq:spin_orbit']}.
  • Figure 5: Isoscalar spin--orbit correlation (a) and quark orbital angular momentum (b) versus $-t$ at $\mu=2$ GeV, computed from Eqs. \ref{['eq:Ji_decomp']} and \ref{['eq:spin_orbit']} at $\eta=0$. Bands: this work (PDF-propagated uncertainties). Points: lattice results Bhattacharya:2024wtgLHPC:2007blg.
  • ...and 1 more figures