Quark Wigner Distributions and Orbital Angular Momentum (1106.0139v1)

Abstract: We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space. These functions can be obtained through a Fourier transform in the transverse space of the generalized transverse-momentum dependent parton distributions. They depend on both the transverse position and the three-momentum of the quark relative to the nucleon, and therefore combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. We focus the discussion on the distributions of unpolarized/longitudinally polarized quark in an unpolarized/longitudinally polarized nucleon. In this way, we can study the role of the orbital angular momentum of the quark in shaping the nucleon and its correlations with the quark and nucleon polarizations. The quark orbital angular momentum is also calculated from its phase-space average weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon. The corresponding results obtained within different light-cone quark models are compared with alternative definitions of the quark orbital angular momentum, as given in terms of generalized parton distributions and transverse-momentum dependent parton distributions.

• The paper introduces a framework that unifies GPDs and TMDs into five-dimensional quark Wigner distributions using a Fourier transform of GTMDs.
• The paper employs light-cone and chiral quark-soliton models to compute quark orbital angular momentum and its influence on nucleon structure.
• The paper compares various OAM definitions, showing agreement in total quark OAM while highlighting differences in flavor-specific contributions.

Quark Wigner Distributions and Orbital Angular Momentum

This paper presents an investigation into Wigner functions for nucleons, which yield multidimensional images of quark distributions in phase space. These functions are derived via a Fourier transform in the transverse space of the Generalized Transverse-Momentum Dependent Parton Distributions (GTMDs). Such Wigner functions depend on the transverse position and the three-momentum of the quark relative to the nucleon, effectively combining all information encapsulated within generalized parton distributions (GPDs) and transverse-momentum dependent parton distributions (TMDs) into a singular comprehensive framework.

The authors focus on unpolarized or longitudinally polarized quarks within an unpolarized or longitudinally polarized nucleon. This approach allows for an examination of the role of the quark's orbital angular momentum (OAM) in shaping the nucleon. The research provides insights into quark orbital angular momentum calculations using phase-space averages weighted by the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon, contrasting these with alternative definitions derived from GPDs and TMDs.

Key Elements and Findings

• Wigner Distributions: Originally explored in QCD for quarks and gluons, Wigner distributions represent the most fundamental parton information in a six-dimensional space. However, this work proposes studying them as five-dimensional distributions in the infinite momentum frame, utilizing two transverse position coordinates and three momentum coordinates.
• Generalized Transverse-Momentum Dependent Parton Distributions: At leading twist, there are 16 GTMDs, accessible through transverse polarizations. The paper specifically addresses the four configurations excluding transverse polarization, leading to a practical understanding by only focusing on longitudinal components.
• Quark Orbital Angular Momentum: This work highlights the complexities of defining quark OAM. The Ji sum rule and its alternative models (like relations to the TMD $h_{1T}^\perp$ and $F_{1,4}$ GTMD) are discussed. Notably, theoretical models, lacking gauge fields, provide consistent results across different definitions for the total quark OAM, though they diverge for specific flavor contributions.
• Phenomenological Models: Employing light-cone constituent quark and chiral quark-soliton models, the paper assesses Wigner distributions through model-specific calculations, yielding distributions exhibiting a substantial influence of quark OAM in nucleon structure. These models capture distinctive multipole structures in distributions, reflecting quark-nucleon polarization correlations.

Implications and Future Direction

The exploration of Wigner distributions provides a richer multidimensional glimpse into nucleon structure, extending beyond traditional GPDs and TMDs. Such distributions open avenues for a deeper understanding of nucleon structure and the intrinsic angular momentum contributions, potentially resolving open issues branded as the "spin crisis."

While experimental access to Wigner distributions remains elusive, a synthesis of evolving theoretical models with experimental data on GPDs and TMDs could yield more comprehensive insights. Models integrating Wigner distributions could thus become pivotal in nuclear physics research, improving our theoretical understanding and paving the way for future explorations in quark dynamics and QCD phenomenology.

The results encapsulate distinct distribution behaviors in phase space for unpolarized vs. polarized configurations, strengthening the narrative that quark OAM and its various correlations substantially shape nucleon structure. Integrating insights from Wigner distributions could also bridge understanding between different theoretical approaches and empirical observations, potentially guiding new measurement techniques and improving existing data interpretation methodologies.

In conclusion, the development and application of Wigner functions in this context represent a critical intersection of theoretical innovation and foundational particle physics inquiry. The nuanced understanding of nucleonic quark distributions they facilitate holds promise for addressing longstanding questions in hadronic physics, particularly concerning the nuanced interplay of spin, momentum, and spatial distributions at quantum levels.