Impact Parameter Dependent Parton Distributions and Transverse Single Spin Asymmetries

TL;DR

The paper shows that generalized parton distributions with transverse momentum transfer provide a tomographic view of partons in the nucleon, where $H_q(x,0,-\mathbf{\Delta}_\perp^2)$ Fourier transforms to the impact-parameter distribution $q(x,\mathbf{b}_\perp)$ and the helicity-flip GPD $E_q(x,0,t)$ induces a transverse distortion in this distribution for a transversely polarized target. This distortion, captured by $q_X(x,\mathbf{b}_\perp)=q(x,\mathbf{b}_\perp) - \frac{1}{2M}\frac{\partial}{\partial b_y}{\cal E}_q(x,\mathbf{b}_\perp)$, is tied to the quark's orbital angular momentum via the Ji sum rule and leads to sizable flavor dipole moments $d^y_q=\frac{1}{2M}\int dx\,E_q(x,0,0)=\frac{F_{2,q}(0)}{2M}$ of order $0.1$–$0.2$ fm. The authors present a crude, yet predictive, model translating these spatial distortions into transverse single-spin and beam-target spin asymmetries through final-state interactions, offering qualitative explanations for observed patterns in hyperon and meson production and highlighting connections to the Sivers mechanism and to spin-transfer observables. The work thus links the internal spin-and-orbital structure of the nucleon to measurable polarization effects in high-energy scattering, with potential implications for tomographic imaging of nucleons and for interpreting spin phenomena in QCD.

Abstract

Generalized parton distributions (GPDs) with purely transverse momentum transfer can be interpreted as Fourier transforms of the distribution of partons in impact parameter space. The helicity-flip GPD $E(x,0,-\Dps)$ is related to the distortion of parton distribution functions in impact parameter space if the target is not a helicity eigenstate, but has some transverse polarization. This transverse distortion can be used to develop an intuitive explanation for various transverse single spin asymmetries.

Figures (7)

• Figure 1: $u$ 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 2: Same as Fig. \ref{['fig:panelu']}, but for $d$ quarks.
• Figure 3: Inclusive $p\longrightarrow Y$ scattering where the incoming $p$ (from bottom) diffractively hits the right side of the target and is therefore, according to the model assumptions, 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 $Y$.
• Figure 4: Schematic view of the transverse distortion of the $s$ quark distribution (in grayscale) in the transverse plane for a transversely polarized hyperon with $\kappa_s^Y>0$. The view is (from the rest frame) into the direction of motion (i.e. momentum into plane) for a hyperon that moves with a large momentum. In the case of spin down (a), the $s$-quarks get distorted toward the left, while the distortion is to the right for the case of spin up (b).
• Figure 5: Transverse polarizations of hyperons that are produced from an unpolarized beam and target (represented by an empty circle). According to the model assumptions, the final state hadron is deflected in the direction given by the side on which the missing quarks were produced. $\odot$ and $\otimes$ represent hyperons with spin pointing out of the plane and into the plane respectively.