Energy-momentum tensor form factors and spin density distribution in the nucleon calculated in a quantized Skyrme model with vector mesons

This paper investigates energy-momentum tensor form factors and spatial spin density distributions in the nucleon using a quantized Skyrme model with vector mesons, demonstrating that while global nucleon properties remain invariant, the choice of pseudogauge (canonical versus Belinfante) significantly alters the local interpretation of spin and momentum densities.

The Gist

Imagine the proton (the core of an atom's nucleus) not as a tiny, solid marble, but as a bustling, three-dimensional city. Inside this city, there are "citizens" (quarks and gluons) zooming around, carrying energy, momentum, and spin. Physicists want to map this city: Where is the pressure highest? How is the "spin" (the city's rotation) distributed?

This paper is like a team of architects using two different blueprints to draw the same city. They discover that while the total size and weight of the city are the same in both blueprints, the local details—like where the streets are or how the wind blows—look completely different depending on which blueprint you use.

Here is a breakdown of their findings using simple analogies:

1. The Two Blueprints: "Canonical" vs. "Belinfante"

In physics, there are two main ways to write down the rules for how energy and momentum flow. The authors call these the Canonical and Belinfante forms.

• The Analogy: Imagine you are describing a spinning ice skater.
  • Blueprint A (Canonical): You describe the spin by saying, "The skater's arms are moving fast, and her body is rotating." You separate the motion of her limbs (spin) from the motion of her body's center (orbit).
  • Blueprint B (Belinfante): You describe the spin by saying, "The entire skater is a single, solid spinning object." You don't separate the arms from the body; you just say the whole thing has angular momentum.

Both descriptions result in the same total spin for the skater. However, if you ask, "How much spin is in the skater's left hand right now?" Blueprint A gives a specific answer, while Blueprint B says, "The spin is distributed everywhere in the body, so the hand doesn't have its own separate spin."

2. The "Pseudogauge" Problem

The paper investigates something called pseudogauge freedom. This is a fancy way of saying: The laws of physics allow us to choose between these two blueprints, and neither is "wrong."

• The Metaphor: Think of a map of a city.
  • Map A draws the city boundaries based on where the water flows.
  • Map B draws the city boundaries based on where the wind blows.
  • Both maps show the same total area of the city (the "global properties" like total mass and total spin).
  • But, if you look at a specific street corner, Map A might say it's a park, while Map B says it's a factory. The local reality looks different depending on which map you hold.

The authors found that in the proton, this choice changes how we see the pressure (how hard the quarks push against each other) and the spin density (where the rotation is happening).

3. The Key Findings

A. The Pressure Map (The "D-term")

The proton is held together by a delicate balance of forces. Some parts push out (like a balloon), and some pull in (like a rubber band).

• The Discovery: When the authors used the "Canonical" blueprint, the pressure map looked one way. When they used the "Belinfante" blueprint, the pressure map looked different.
• The Takeaway: If you ask a physicist, "What is the pressure at the center of the proton?" the answer depends on which "map" (pseudogauge) they are using. This means our current experimental data (which mostly sees the "total" picture) isn't detailed enough to tell us which map is the "true" local reality.

B. The Spin Distribution (The "J-term")

This is about where the proton's spin comes from.

• The Canonical View: The spin is a mix of two things: the "orbit" of the quarks moving around, and their intrinsic "spin" (like a tiny top spinning). In this view, the vector mesons (heavy particles inside the proton) act like tiny tops that contribute their own spin.
• The Belinfante View: In this view, the "intrinsic spin" of those tiny tops is mathematically absorbed into the "orbit." It looks like the whole proton is just orbiting, and there is no separate "spin" component left over.
• The Result: The authors calculated that in the Canonical view, about 36% of the spin comes from orbit and 14% comes from intrinsic spin (the rest is other stuff). In the Belinfante view, it looks like 100% is orbital. The total is the same (50%), but the breakdown is totally different.

4. Why Does This Matter?

You might ask, "If the total spin is the same, why do we care?"

• The "City Planning" Analogy: If you are a city planner trying to fix traffic, it matters where the traffic jams are. If you think the jam is in the downtown park (Canonical view) but it's actually in the industrial zone (Belinfante view), you will build the wrong road.
• The Future: The upcoming Electron-Ion Collider (EIC) is like a super-powerful camera that will take 3D pictures of the proton. This paper warns us: Be careful how you interpret the photos. The camera might show a specific pattern of pressure or spin, but without knowing which "blueprint" nature is using, we might misinterpret the internal mechanics of the proton.

Summary

This paper is a "reality check" for physicists. It shows that while we know the total weight and spin of a proton, our understanding of where that weight and spin are located inside the proton depends on a mathematical choice we make.

It's like looking at a spinning galaxy through two different colored glasses. The galaxy spins the same speed either way, but one glass makes the stars look like they are orbiting the center, while the other makes them look like they are spinning in place. Both are valid descriptions, but they tell very different stories about the galaxy's internal structure. The authors have built a model to visualize these two stories side-by-side, helping us prepare for the new, high-definition data coming from the EIC.

Technical Summary

Here is a detailed technical summary of the paper "Energy-momentum tensor form factors and spin density distribution in the nucleon calculated in a quantized Skyrme model with vector mesons" by Kenji Fukushima and Tomoya Uji.

1. Problem Statement

The paper addresses the pseudogauge ambiguity in the definition of the Energy-Momentum Tensor (EMT) within Quantum Chromodynamics (QCD) and effective field theories.

• The Core Issue: While the total conserved charges (momentum and angular momentum) are unique, the local densities of momentum and spin are not. Different "pseudogauges" (specifically the Canonical vs. Belinfante constructions) yield different spatial distributions for these quantities, even though they agree on global integrals.
• The Gap: Current experimental programs, such as the Electron-Ion Collider (EIC), aim to map the 3D structure of the nucleon via Generalized Parton Distributions (GPDs). However, leading-twist observables (accessible via Deeply Virtual Compton Scattering) primarily probe the symmetric, light-front projected sector of the EMT, effectively masking the differences between pseudogauges. There is a lack of concrete models that explicitly visualize how these different definitions alter the spatial interpretation of nucleon spin and mechanical properties (pressure/shear forces).
• Specific Challenge: In gauge theories, separating intrinsic spin and orbital angular momentum (OAM) is subtle. The paper seeks to quantify how the choice of EMT definition affects the decomposition of total angular momentum into spin and orbital components at the density level.

2. Methodology

The authors employ a quantized Skyrme model extended with vector mesons ($\rho$ and $\omega$) as a theoretical laboratory.

• Model Framework:

  • Lagrangian: Uses a chiral effective Lagrangian including pion fields ($U$), isotriplet vector mesons ($\rho_\mu$), and an isosinglet vector meson ($\omega_\mu$).
  • Soliton Solution: A static "hedgehog" soliton solution is derived for the baryon number $B=1$.
  • Collective Quantization: To describe nucleon states with definite spin ($J=1/2$), the model introduces adiabatic collective rotations. This induces time-dependent vector meson components and generates non-zero momentum densities.
  • Quantization: Rotational zero modes are quantized, promoting the angular velocity to the spin operator $\hat{J}$.

• EMT Construction:

  • Canonical EMT ($T^{\mu\nu}_{can}$): Derived via Noether's theorem. It is generally non-symmetric ($T^{0i} \neq T^{i0}$) and contains an explicit spin current tensor $S^{\lambda\mu\nu}$.
  • Belinfante EMT ($T^{\mu\nu}_{Bel}$): Derived via metric variation (Hilbert definition) or by adding a superpotential term to the canonical tensor. It is symmetric ($T^{\mu\nu} = T^{\nu\mu}$) and absorbs the spin current into the orbital part.

• Analysis Techniques:

  • Form Factors: The authors calculate nucleon matrix elements in the Breit frame ($\Delta^0=0$) to extract form factors: $A(t)$ (momentum), $D(t)$ (mechanical/pressure), $J(t)$ (total angular momentum), and $G(t)$ (antisymmetric form factor unique to the canonical tensor).
  • Inversion Formulas: They utilize Fourier-Bessel transformations to convert the calculated spatial densities (in coordinate space) back into momentum-transfer dependent form factors, and vice versa, to visualize 3D distributions.
  • Decomposition: They explicitly separate the total angular momentum density into orbital ($L$) and spin ($S$) contributions for both pseudogauges.

3. Key Contributions

1. Explicit Model Realization: This is one of the first studies to construct both canonical and Belinfante EMTs within the same dynamical framework (Skyrme model with vector mesons) to directly compare their local densities.
2. Visualization of Pseudogauge Dependence: The paper provides explicit 3D visualizations showing that while global charges are conserved, the local spatial distribution of spin and momentum differs drastically between the two definitions.
3. Role of Vector Mesons: The inclusion of vector mesons is crucial. In a pion-only Skyrme model, the intrinsic spin density is zero. The vector meson sector provides a non-zero intrinsic spin density in the canonical decomposition, allowing for a meaningful separation of $L$ and $S$.
4. Clarification of $G(t)$: The authors extract the antisymmetric form factor $G(t)$, which vanishes in the Belinfante limit but is non-zero in the canonical limit. They clarify its relationship to the orbital angular momentum charge.

4. Key Results

A. Form Factors

• Momentum Form Factor $A(t)$: Both constructions satisfy the normalization $A(0)=1$, consistent with translational invariance.
• Mechanical Form Factor $D(t)$:
  • Significant difference found in the forward limit: $D_{can}(0) \approx -1.30$ vs. $D_{Bel}(0) \approx -3.88$.
  • Implication: The "pressure" and "shear force" distributions inferred from $D(t)$ are not unique. Leading-twist GPD data cannot distinguish which pseudogauge underlies the physical pressure distribution, introducing a systematic uncertainty in mechanical interpretations.
• Angular Momentum Form Factor $J(t)$:
  • Belinfante: $J_{Bel}(0) = 1/2$, satisfying the spin-1/2 normalization directly from the symmetric tensor.
  • Canonical: $J_{can}(0) \neq 1/2$. The quantity extracted from the symmetric part of the canonical tensor does not equal the total spin.
  • Restoration: The total angular momentum is recovered only when adding the antisymmetric contribution: $J_{total} = J_{can}(0) + \frac{1}{2}G_{can}(0) = 1/2$.

B. Spatial Distributions (Spin and Momentum)

• Canonical Decomposition:
  • The total angular momentum is split into a non-zero orbital part ($L_{can} \approx 0.36$) and a non-zero intrinsic spin part ($S_{can} \approx 0.14$).
  • The spin density $S_z^{can}$ is finite even at the center of the nucleon ($r=0$).
• Belinfante Decomposition:
  • The spin current is absorbed into the EMT. Consequently, the "spin density" is zero ($S_{Bel}=0$).
  • The total angular momentum is encoded entirely in an "orbital-like" term ($x^i T^{0j} - x^j T^{0i}$).
  • Crucial Finding: The Belinfante angular momentum density vanishes at the center of the nucleon ($r=0$), whereas the canonical spin density remains finite there. This highlights that the "location" of spin depends entirely on the pseudogauge choice.

C. Boost Densities

• The authors analyzed boost-related densities ($L^{00i}$ and $S^{00i}$). In the nucleon rest frame, the total boost charge vanishes for both constructions, consistent with the nucleon being at rest.

5. Significance and Implications

• Interpretation of EIC Data: The results warn that interpreting 3D nucleon tomography (pressure, shear, spin density) requires acknowledging the pseudogauge ambiguity. Leading-twist experiments (DVCS) constrain the symmetric sector but cannot resolve the local spin distribution's definition.
• Beyond Leading Twist: To experimentally distinguish between these definitions, one must access higher-twist observables (e.g., higher-twist GPDs, GTMDs, or TMDs) where the process-dependent Wilson lines (which define the gauge-invariant extension) become essential.
• Mechanical Stability: The large discrepancy in $D(0)$ implies that theoretical predictions for the internal pressure of the nucleon have an inherent model-dependent ambiguity regarding the choice of EMT definition.
• Spin Decomposition: The study demonstrates that the separation of spin into "intrinsic" and "orbital" components is not a physical observable in isolation but depends on the chosen pseudogauge. The "spin" is not a localized object in the same way in the Belinfante picture as it is in the Canonical picture.

Conclusion:
The paper provides a concrete, quantitative demonstration that the spatial distribution of nucleon spin and mechanical forces is not unique but depends on the pseudogauge choice. While global properties remain invariant, the local interpretation of "where the spin is" and "how the pressure is distributed" varies significantly between the Canonical and Belinfante frameworks. This underscores the need for caution when mapping theoretical densities to experimental observables and highlights the necessity of higher-twist measurements to fully resolve the nucleon's internal structure.