Transverse energy-momentum tensor distributions in polarized nucleons

This paper completes the study of relativistic spatial distributions of the energy-momentum tensor in polarized nucleons by focusing on transverse components and their multipole structure, demonstrating that the quantum phase-space formalism reproduces standard light-front distributions in the infinite-momentum frame.

The Gist

Imagine a proton (a nucleon) not as a tiny, hard marble, but as a bustling, chaotic city made of smaller particles called quarks and gluons. Inside this city, there is constant motion: particles zooming around, colliding, and pushing against each other. This paper is about mapping out exactly how this city is built, how it holds itself together, and how it spins.

Specifically, the authors are looking at the Energy-Momentum Tensor (EMT). In physics terms, this is a complex mathematical object that tells us about the energy, pressure, and forces inside the proton. But let's translate that into everyday language.

The Big Challenge: Taking a Photo of a Moving City

The main problem the scientists are solving is: How do you take a clear picture of a city that is moving at nearly the speed of light?

1. The "Rest Frame" Problem: If you try to take a picture of the proton while it's sitting still (or moving very slowly), the laws of quantum mechanics and relativity get messy. It's like trying to photograph a spinning top with a slow shutter speed; the image blurs. The proton is so small that its "size" is comparable to the wavelength of the light used to see it, making a simple snapshot impossible.
2. The "Fast Frame" Problem: If you watch the proton zooming past at near light-speed (the "Infinite Momentum Frame"), the picture changes again. The city looks flattened and distorted, like a pancake.

The authors use a clever mathematical tool called the Quantum Phase-Space Formalism. Think of this as a super-camera that can take a picture of the proton at any speed, and then mathematically "stitch" those different views together to create a consistent, 3D map of the internal forces.

The New Discovery: The "Sideways" Forces

In previous studies, the authors only looked at forces pushing straight forward or backward (longitudinal). In this paper, they finally looked at the sideways (transverse) forces.

Imagine the proton city again.

• Longitudinal forces are like traffic flowing down a highway.
• Transverse forces are like the wind blowing sideways, or the pressure of a crowd pushing against the side of a building.

The authors mapped out these sideways pressures and flows. They found that:

• Spin Matters: If the proton is spinning (polarized), it creates a "whirlpool" effect. The sideways forces aren't just random; they swirl around the center, creating a specific pattern of stress.
• The "Wigner Rotation": This is a fancy term for a subtle twist. When you look at a spinning object while moving past it at high speed, the object appears to tilt or rotate slightly relative to your view. The authors showed how this "tilt" distorts the map of the internal forces, but only when the proton is moving.

The "Perfect" View: The Infinite Momentum Frame

One of the coolest findings is what happens when the proton moves infinitely fast (the "Infinite Momentum Frame").

The authors showed that their complex, speed-dependent maps simplify perfectly in this extreme case. The messy distortions caused by the proton's speed disappear, and the map becomes identical to the standard "Light-Front" distributions used by physicists today.

The Analogy:
Imagine looking at a spinning fan.

• When the fan is slow, it looks like a blur, and it's hard to tell where the blades are.
• As you speed up your own movement relative to the fan, the blur changes shape.
• But if you move with the fan at the exact same speed, or if you look at it from a specific angle where the physics simplifies, the blades suddenly snap into a clear, sharp image.

This paper proves that the "blurry" relativistic maps and the "sharp" light-speed maps are actually two sides of the same coin. The authors provided the mathematical bridge that explains why they look different at different speeds and how they connect.

Why Does This Matter?

Understanding these internal forces is crucial for two big reasons:

1. The Origin of Mass: Most of the proton's mass doesn't come from the weight of its parts, but from the energy of their motion and the pressure holding them together. This map helps us understand where that mass comes from.
2. The Future of Science: The upcoming Electron-Ion Collider (EIC) will act like a giant, high-speed camera to take real pictures of these internal structures. This paper provides the theoretical "instruction manual" for how to interpret those future photos.

Summary

In short, this paper is a 3D atlas of the forces inside a spinning proton.

• It maps out the sideways winds and pressures (transverse energy-momentum) that were previously ignored.
• It explains how these forces twist and distort depending on how fast the proton is moving.
• It proves that when you zoom in at the highest possible speeds, these complex maps simplify into the standard pictures physicists have been using for years, confirming that our understanding is consistent.

It's a bit like realizing that a spinning top looks different from the side than it does from the front, but with the right math, you can predict exactly how it will look from any angle, no matter how fast it's spinning.

Technical Summary

1. Problem Statement

The internal structure of nucleons, particularly the spatial distribution of the energy-momentum tensor (EMT), is a central challenge in hadron physics. While previous studies have mapped EMT distributions in unpolarized nucleons or focused on specific components (those without transverse indices) in polarized nucleons, a complete relativistic description of the transverse components of the EMT in polarized nucleons was lacking.

Specifically, the authors address:

• The relativistic spatial distributions of EMT components involving at least one transverse index ($T^{0i}, T^{i0}, T^{3i}, T^{i3}, T^{ij}$) for polarized nucleons.
• The multipole structure of these distributions in moving frames.
• The reconciliation of these distributions with standard Light-Front (LF) distributions in the infinite-momentum frame (IMF), including components often considered "bad" (non-dynamical) in other formalisms.

2. Methodology

The authors utilize the Quantum Phase-Space Formalism, which allows for the definition of relativistic spatial distributions in frames where the nucleon has a non-vanishing average momentum (Elastic Frames, EF), interpolating between the Breit Frame (BF) and the Infinite-Momentum Frame (IMF).

Key Technical Steps:

• Operator Definition: They adopt the asymmetric, gauge-invariant local EMT operator for quarks and gluons in QCD.
• Matrix Element Parametrization: The EMT matrix elements are parameterized using multipole form factors (FFs) ($E, J, F_0, F_2, S$) for a spin-1/2 state.
• Frame Transformation: They analyze the transformation of these matrix elements from the Transverse Breit Frame (where $\Delta_z = 0, P_z = 0$) to a general Elastic Frame (where $\Delta_z = 0, P_z \neq 0$) using Lorentz boosts and Wigner spin rotations.
• Fourier Transform: The 2D relativistic spatial distributions in the transverse plane ($b_\perp$) are defined via the Fourier transform of the EF matrix elements.
• Normalization: To handle the divergence of energy in the IMF, they define normalized EMT distributions by factoring out Lorentz boost factors ($\gamma_P$).
• Multipole Decomposition: The distributions are decomposed into irreducible symmetric-traceless tensors (monopole, dipole, quadrupole, etc.) to reveal their angular structure.
• Light-Front Comparison: They derive the corresponding Light-Front amplitudes in the Drell-Yan frame and compare them with the IMF limit of their relativistic distributions.

3. Key Contributions and Results

A. Vector Distributions (Transverse Momentum and Flux)

The authors analyze components $T^{0i}$ (transverse momentum density), $T^{3i}$ (longitudinal flux of transverse momentum), $T^{i0}$ (transverse energy flux), and $T^{i3}$ (transverse flux of longitudinal momentum).

• Pure Dipolar Nature: These components are purely dipolar.
• Momentum Independence: The transverse momentum distribution ($P_\perp$) is independent of the nucleon's longitudinal momentum $P_z$. It depends only on the difference between the total angular momentum form factor $J(Q^2)$ and the intrinsic spin form factor $S(Q^2)$.
• Stress Distortion: The longitudinal flux of transverse momentum (LT stress, $\Pi^{zi}_\perp$) and transverse flux of longitudinal momentum (TL stress, $\Pi^{iz}_\perp$) depend on $P_z$. They are distorted by a relativistic factor involving $\gamma_P/\gamma$.
• IMF Limit: In the infinite-momentum frame ($P_z \to \infty$), the distortions vanish, and the LT/TL stresses become identical to the transverse momentum and transverse energy flux distributions, respectively.
• Angular Momentum: The orbital angular momentum (OAM) and intrinsic spin are recovered from the moments of these distributions: $L_z \propto J-S$ and $S_z \propto S$.

B. Tensor Distributions (Transverse Stress)

The authors analyze the transverse stress tensor $T^{ij}$, decomposing it into isotropic ($\sigma$) and anisotropic ($\Sigma_{ij}$) parts.

• Isotropic Stress ($\sigma$):
  • Unlike vector components, these do not mix under longitudinal boosts but undergo Wigner spin rotation.
  • For a transversely polarized nucleon, the isotropic stress distribution exhibits a dipolar distortion perpendicular to the polarization axis.
  • The total integral of the isotropic stress vanishes, satisfying the 2D version of the von Laue condition (global equilibrium).
  • In the model used, the quark contribution dominates the isotropic stress, while gluons largely cancel out due to sign differences in their form factors.
• Anisotropic Stress ($\Sigma_{ij}$):
  • This distribution also undergoes Wigner rotation, leading to a dipolar shift in transversely polarized states.
  • The anisotropic stress is dominated by gluons in the multipole model used, as $|D_G| > |D_q|$.
  • The quadrupole moment of the anisotropic stress is independent of $P_z$ and nucleon polarization, depending only on the form factor $F_2(0)$.

C. Connection to Light-Front (LF) Distributions

• The authors demonstrate that the relativistic spatial distributions derived in the Elastic Frame converge to the standard Light-Front distributions in the IMF limit ($P_z \to \infty$).
• They explicitly show that the "bad" components (those involving $T^{+-}$ or similar in other contexts, here represented by the specific transverse indices in the LF formalism) are naturally reproduced by the formalism.
• The equivalence is established via the Melosh rotation, which relates canonical polarization to LF helicity. In the IMF, the Melosh rotation becomes the identity, explaining why the EF and LF results coincide.

4. Significance

• Completeness: This work completes the relativistic spatial mapping of the EMT for polarized nucleons, covering all tensor components, not just the scalar ones.
• Frame Independence vs. Dependence: It clarifies which EMT components are frame-independent (transverse momentum) and which are frame-dependent (stresses), providing a unified picture of how these quantities evolve from the rest frame to the IMF.
• Physical Interpretation: The study provides a rigorous derivation of the origin of relativistic distortions (Wigner rotation and Lorentz contraction) in spatial distributions. It shows that these distortions are artifacts of the frame choice and disappear in the IMF, where the distributions acquire a probabilistic density interpretation.
• Experimental Relevance: These results are crucial for the upcoming Electron-Ion Collider (EIC), which aims to map the 3D structure of nucleons. The paper provides the theoretical framework to interpret future measurements of EMT form factors and their spatial distributions.
• Spin-Orbit Correlations: By analyzing the parity-odd partners and the specific structure of the transverse distributions, the work offers insights into spin-orbit correlations and the chiral structure of hadrons.

In summary, Won and Lorcé provide a comprehensive theoretical framework that bridges the gap between Breit frame intuition and Light-Front dynamics, offering a complete picture of the mechanical and angular momentum properties of polarized nucleons.