T-odd Wigner Distributions in boost-invariant longitudinal position space and Spin-momentum correlation in proton

This paper investigates the skewness sensitivity of T-odd Generalized Transverse Momentum Dependent distributions (GTMDs) in boost-invariant longitudinal position space, revealing that their oscillatory diffraction patterns are influenced by momentum transfer and interference effects, while also exploring spin-momentum correlations through Sivers and Boer-Mulders Wigner distributions.

The Gist

Imagine you are trying to understand the inner workings of a high-speed, spinning top—but this top is a proton (one of the tiny building blocks of atoms), and it is spinning so fast and is so small that you can’t just look at it with a microscope. Instead, you have to use complex math and high-energy particle collisions to "see" what’s happening inside.

This paper is essentially a high-tech "3D mapping" project of the proton's interior. Here is the breakdown of what the researchers did, using some everyday analogies.

1. The Goal: Creating a 3D Hologram

Normally, when scientists look at a proton, they get a "flat" picture—like a 2D photograph. But a proton isn't flat; it’s a swirling storm of quarks and gluons moving in all directions.

The researchers are using something called Wigner Distributions. Think of this as a 3D holographic map. Instead of just knowing where a particle is, this map tells you both where it is (position) and where it’s going (momentum) at the exact same time.

2. The "Skewness" Problem: The Moving Target

In most studies, scientists assume the proton is being hit "straight on." But in the real world, when you hit a proton with energy, the proton often recoils or shifts slightly. This shift is called skewness.

Imagine you are trying to take a photo of a person running on a treadmill. If they are running perfectly straight, it’s easy. But if they are zig-zagging or shifting side-to-side while running, the photo gets blurry and complicated. The researchers are specifically looking at how this "zig-zagging" (skewness) changes the internal map of the proton.

3. The "Sivers" and "Boer-Mulders" Effects: The Swirl in the Storm

The paper focuses on two specific phenomena: the Sivers effect and the Boer-Mulders effect.

• The Sivers Effect: Imagine a whirlpool in a river. If you throw a leaf into the whirlpool, the leaf doesn't just go straight; it gets pushed to one side because of the spin of the water. The Sivers effect is like that—it describes how the spin of the proton pushes the quarks to one side, creating an imbalance.
• The Boer-Mulders Effect: This is similar, but instead of the whole proton spinning, it’s about how the individual quarks are spinning and how that affects their movement.

4. The Big Discovery: The "Optical Diffraction" Pattern

The most exciting part of the paper is what happens when they look at the proton in "longitudinal position space" (a specific way of measuring distance inside the proton).

They found that the internal distributions of these quarks show oscillating patterns. This is exactly like light passing through a tiny slit (what scientists call "diffraction").

The Analogy: Imagine shining a laser pointer through a very thin piece of paper with a tiny slit cut in it. On the wall, you don't just see one dot of light; you see a pattern of bright and dark stripes. The researchers found that the quarks inside the proton behave like those light waves. By studying these "stripes" (the patterns of the quarks), they can work backward to figure out exactly how "wide the slit" is—or, in physics terms, how the momentum is being transferred during a collision.

Summary: Why does this matter?

By creating these incredibly detailed, 3D, "holographic" maps that account for the proton's spin and its "zig-zagging" movement, scientists are getting closer to a complete "weather report" for the inside of an atom.

Understanding this "storm" of quarks is fundamental to understanding how all matter in the universe is held together. It’s like moving from a blurry, black-and-white sketch of a city to a high-definition, 3D GPS map that shows not just where the buildings are, but how the traffic is flowing through every single street.

Technical Summary

Technical Summary: T-odd Wigner Distributions in Boost-Invariant Longitudinal Position Space and Spin-Momentum Correlation in Proton

1. Problem Statement

The internal structure of the proton is characterized by the complex correlations between the spatial and momentum distributions of its constituents (quarks and gluons). While Generalized Parton Distributions (GPDs) and Transverse Momentum Distributions (TMDs) provide partial "tomographic" views, the most comprehensive description is offered by Wigner Distributions (WDs) and Generalized Transverse Momentum Distributions (GTMDs).

A significant gap exists in the current understanding of the T-odd (Time-reversal odd) sector of these distributions—specifically those responsible for Single-Spin Asymmetries (SSAs) like the Sivers and Boer-Mulders effects—when the skewness parameter ($\xi$) is non-zero. Most existing models assume $\xi = 0$ (momentum transfer purely in the transverse direction), but experimental realities (such as in Deeply Virtual Compton Scattering) involve longitudinal momentum transfer ($\xi \neq 0$). Furthermore, the representation of these distributions in boost-invariant longitudinal position space ($\sigma$) remains largely unexplored.

2. Methodology

The author employs the Light-Front Quark-Diquark Model (LFQDM), an effective framework where the proton is treated as a bound state of a quark and a diquark (both scalar and axial-vector).

• Wave Functions: The Light-Front Wave Functions (LFWFs) are derived from the soft-wall AdS/QCD framework, ensuring a connection to non-perturbative QCD dynamics.
• T-odd Mechanism: To account for the T-odd sector, the model incorporates Final State Interactions (FSI) by introducing an imaginary part into the LFWFs via gluon exchange.
• Kinematics: The study operates within the DGLAP region ($\xi < x < 1$). The author utilizes a symmetric frame to define the kinematics of the momentum transfer $\Delta$ and the skewness $\xi$.
• Mathematical Transformations:
  • GTMDs to WDs: The author performs Fourier transforms of the GTMDs with respect to the skewness $\xi$ to obtain distributions in the boost-invariant longitudinal position space $\sigma$.
  • Impact Parameter Space: Fourier transforms with respect to transverse momentum transfer $\Delta_\perp$ are used to derive distributions in the transverse impact parameter space $b_\perp$.

3. Key Contributions

• Extension to Non-Zero Skewness: The paper provides the first systematic investigation of leading-twist T-odd GTMDs ($F_{1,2}^{(o)}$ and $H_{1,1}^{(o)}$) with non-zero skewness.
• Longitudinal Tomography: It introduces the study of Sivers and Boer-Mulders Wigner distributions in the boost-invariant longitudinal position space ($\sigma$), providing a new dimension for proton tomography.
• Interference Analysis: The work identifies and analyzes the interference effects between the transverse momentum of the quark ($p_\perp$) and the transverse momentum transfer ($\Delta_\perp$).
• Spin-Momentum Correlation: The paper quantifies how skewness affects the spin density and the mean-square transverse momentum $\langle p_\perp^2 \rangle$ of quarks.

4. Key Results

• GTMD Behavior:
  • The T-odd GTMD $F_{1,2}^{(o)}$ (associated with the Sivers effect) shows a polarity flip between $u$ and $d$ quarks, which is a prerequisite for the Sivers asymmetry.
  • The $H_{1,1}^{(o)}$ distribution (Boer-Mulders) remains positive for both flavors.
  • As $\xi$ increases, the magnitudes of these distributions decrease, and their peaks shift toward higher $x$.
• Diffraction Patterns in $\sigma$-space:
  • The Wigner distributions in longitudinal position space exhibit oscillatory patterns analogous to optical diffraction.
  • The width of the central maxima is sensitive to the total momentum transfer $-t$; as $-t$ increases, the diffraction pattern becomes narrower (resembling a narrower slit in optics).
  • Interference Effect: When $p_\perp$ and $\Delta_\perp$ are parallel, the diffraction pattern becomes asymmetric. When they are perpendicular, the pattern becomes symmetric, resembling single-slit diffraction.
• Spin Density and Asymmetry:
  • The spin density of unpolarized quarks in a transversely polarized proton shows a distinct left-right shift for $u$ and $d$ quarks, directly linked to the Sivers effect.
  • The mean-square transverse momentum $\langle p_\perp^2 \rangle$ decreases as skewness $\xi$ increases.
• Impact Parameter Space: In the $b_\perp$-plane, the distributions exhibit axial symmetry rather than circular symmetry due to the $p_\perp \cdot \Delta_\perp$ interference terms.

5. Significance

This research significantly advances the theoretical landscape of hadron structure. By moving beyond the $\xi = 0$ approximation, it provides a more realistic framework for interpreting upcoming high-precision data from facilities like the Electron-Ion Collider (EIC). The discovery of "diffraction-like" patterns in longitudinal position space offers a novel way to visualize the spatial distribution of partons and provides a new mathematical tool (the $\sigma$ variable) to study the relativistic, boost-invariant properties of the proton.