Pseudogauge ambiguity in the distributions of energy density, pressure, and shear force inside the nucleon

This paper investigates the spatial distributions of energy density, pressure, and shear forces within a two-flavor Skyrme model including vector mesons, revealing that the energy-momentum tensor's pseudo-gauge dependence—arising from nonvanishing surface terms of spin currents—causes significant differences between canonical and Belinfante forms, including singularities in the canonical pressure and shear-force distributions at the nucleon center.

The Gist

Imagine the proton (the core of a hydrogen atom, and a building block of everything around us) not as a tiny, solid marble, but as a bustling, three-dimensional city. Inside this city, there are forces pushing outward and forces pulling inward, all trying to keep the city from falling apart.

For decades, physicists have been trying to map this city: Where is the pressure highest? How strong is the "glue" holding it together? How much energy is packed into the center?

This paper by Fukushima and Uji is like a detective story about how we measure these forces. The authors discovered that the answer depends entirely on which ruler you choose to use.

Here is the breakdown of their discovery using simple analogies.

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

In physics, to measure the energy and pressure inside a particle, we use a mathematical tool called the Energy-Momentum Tensor (EMT). Think of this as a blueprint for the city's forces.

The problem is, there are two different blueprints (or "gauges") that are both mathematically correct:

• The Canonical Blueprint: This is the "raw" version. It's like looking at the city through a slightly distorted, wide-angle lens. It includes some extra "spin" details that make the math work but can look messy.
• The Belinfante Blueprint: This is the "refined" version. It's like looking through a standard, straight-on lens. It smooths out the distortions and is often considered the "standard" way to measure things.

Usually, physicists assume these two blueprints would give the same picture of the city, just with minor cosmetic differences. This paper proves that is wrong.

2. The Shocking Discovery: The Center of the City

The authors used a specific model (the Skyrme model with vector mesons) to simulate the proton. They applied both blueprints to see what the pressure and energy looked like at the very center of the proton.

• The Belinfante View (The Smooth Lens): The pressure at the center is high, but it's a nice, finite number. It's like a skyscraper with a solid, heavy foundation.
• The Canonical View (The Distorted Lens): The pressure at the center blows up to infinity. It's as if the blueprint says the center of the proton is a black hole where the pressure is infinite.

The Analogy: Imagine you are measuring the temperature in the center of a campfire.

• One thermometer (Belinfante) says it's a scorching 1,000°C.
• The other thermometer (Canonical) says it's "Infinity."
• Both thermometers are calibrated correctly according to their own rules, but they tell you completely different stories about the fire's core.

3. Why Does This Happen? (The "Spin" Current)

Why do the two rulers disagree? The authors found that the difference comes from spin currents.

Think of the proton's interior as a spinning top. The "vector mesons" (particles inside the proton) create a swirling magnetic field, like a tiny tornado.

• The Canonical ruler counts the "twist" of this tornado as part of the pressure. Because the twist is so intense at the very center, it creates a mathematical singularity (an infinity).
• The Belinfante ruler decides to ignore that specific twist when calculating pressure, smoothing it out so the numbers stay finite.

The authors realized that the "twist" creates a surface term—a mathematical artifact that changes the local numbers but doesn't change the total weight of the proton.

4. The Big Consequences

This isn't just a math game; it changes how we understand the universe.

• The "Equation of State" (The Recipe): If you want to know how matter behaves under extreme pressure (like inside a neutron star), you need a recipe called the Equation of State (EoS). This paper shows that if you use the "Canonical" ruler, your recipe says the matter is infinitely dense at the center. If you use the "Belinfante" ruler, it says it's just very dense.

  • Result: The "Belinfante" recipe actually matches what we see in real neutron stars much better. The "Canonical" one looks physically impossible.

• The "Confining Force" (The Glue): We know protons are held together by a force that keeps quarks from escaping. The paper shows that the "Canonical" ruler suggests this glue is infinitely strong at the center (to balance the infinite pressure), which seems physically absurd. The "Belinfante" ruler suggests a strong, but manageable, glue.

5. The Bottom Line: Which Ruler is Right?

The authors conclude that there is no single "true" ruler.

• Global properties (like the total mass of the proton or the total stability) are the same no matter which ruler you use. The total weight of the city is the same, even if the blueprint of the center looks different.
• Local properties (what is happening right here at the center) are ambiguous. They depend on which mathematical definition you choose.

The Takeaway:
If you ask a physicist, "What is the pressure at the center of a proton?" the honest answer is: "It depends on which mathematical definition of pressure you are using."

This paper warns scientists that when they try to map the internal structure of protons using data from particle colliders (like the upcoming Electron-Ion Collider), they must be very careful. They cannot assume the "picture" they get is the absolute, unique truth. They have to acknowledge that the "lens" they use to view the proton changes the image they see.

In short: The proton is a stable, heavy city, but our maps of its downtown district are blurry and depend entirely on which mapmaker you ask.

Technical Summary

Here is a detailed technical summary of the paper "Pseudogauge ambiguity in the distributions of energy density, pressure, and shear force inside the nucleon" by Kenji Fukushima and Tomoya Uji.

1. Problem Statement

The internal mechanical structure of the nucleon (mass, spin, and force distributions) is encoded in the matrix elements of the Energy-Momentum Tensor (EMT). These properties are often extracted from Generalized Parton Distributions (GPDs) via hard exclusive processes like Deeply Virtual Compton Scattering (DVCS).

A fundamental theoretical issue arises because the EMT is not uniquely defined. One can perform a pseudo-gauge transformation (adding a total derivative term) to the EMT without violating the conservation law ($\partial_\mu T^{\mu\nu} = 0$). This leads to two common forms:

• Canonical EMT: Derived directly from Noether's theorem. It is generally asymmetric and not gauge invariant in gauge theories but is crucial for spin physics (e.g., Jaffe-Manohar decomposition).
• Belinfante EMT: A symmetrized version obtained via a pseudo-gauge transformation involving the spin current. It is symmetric and gauge invariant, often preferred for gravitational coupling and hydrodynamics.

While global quantities (like total mass) are invariant under these transformations, the local distributions of energy density, pressure, and shear force may differ. The paper addresses the critical question: Do these local mechanical properties depend on the choice of pseudo-gauge, and if so, how does this affect physical interpretations like the confining force and the Equation of State (EoS)?

2. Methodology

The authors utilize the two-flavor Skyrme model extended with vector mesons ($\rho$ and $\omega$) to model the nucleon as a topological soliton.

• Model Selection: Unlike the ordinary Skyrme model, this framework includes vector mesons as dynamical fields. This is crucial because the difference between the Canonical and Belinfante EMTs arises from the spin current, which is non-zero only when vector meson field strength tensors are present.
• Lagrangian: The model uses a Lagrangian density containing pion fields ($U$), $\rho$ mesons, and $\omega$ mesons. The baryon number is stabilized without the Skyrme term due to the presence of vector mesons.
• Ansätze:
  • Pions: Hedgehog Ansatz.
  • $\rho$ mesons: Wu-Yang-'t Hooft-Polyakov Ansatz.
  • $\omega$ mesons: Static, spherically symmetric Ansatz.
• Calculations:
  1. Solve the coupled equations of motion numerically to obtain field profiles ($F(r), G(r), \omega(r)$).
  2. Derive analytical expressions for the energy density ($\varepsilon$), pressure ($p$), and shear force ($s$) for both the Canonical and Belinfante EMTs.
  3. Perform numerical integration to compare the radial distributions of these quantities.
  4. Analyze physical implications, specifically the confining force and the Equation of State (EoS) of single-nucleon matter.

3. Key Contributions

• Explicit Demonstration of Ambiguity: The authors explicitly calculate and demonstrate that local distributions of energy, pressure, and shear force differ significantly between the Canonical and Belinfante forms within a realistic nucleon model.
• Origin Identification: They identify the source of the ambiguity as non-vanishing surface terms associated with spin currents generated by the vector-meson field strength tensors ($F_{\mu\nu}$).
• Singularity Discovery: A major finding is that the Canonical pressure develops a singularity ($p \sim r^{-2}$) at the nucleon center ($r \to 0$), whereas the Belinfante pressure remains finite ($p \sim r^0$).
• Re-evaluation of Mechanical Forces: The study challenges the direct interpretation of negative pressure as the confining force, showing that the "confining force" derived from the shear force is also pseudo-gauge dependent.

4. Key Results

A. Energy Density ($\varepsilon$)

• The local profiles of $\varepsilon_{\text{can}}$ and $\varepsilon_{\text{Bel}}$ differ. $\varepsilon_{\text{can}}$ is more localized near the center.
• However, the total energy (nucleon mass) obtained by integrating $\varepsilon(r)$ is pseudo-gauge invariant, as the difference is a total derivative (surface term) that vanishes at infinity.

B. Pressure ($p$) and Shear Force ($s$)

• Qualitative Difference: The pressure distributions differ drastically.
  • Canonical: $p_{\text{can}}(r) \sim r^{-2}$ as $r \to 0$. This singularity causes the pressure sum rules (beyond the standard von Laue condition) to depend on the pseudo-gauge choice.
  • Belinfante: $p_{\text{Bel}}(r)$ is finite at the origin.
• Shear Force: While the volume integral of the shear force tensor is zero (due to symmetry), the radial integral (often interpreted as "surface tension energy") is not invariant.
  • $\int 4\pi r^2 s_{\text{can}} dr \approx 2.3$ GeV.
  • $\int 4\pi r^2 s_{\text{Bel}} dr \approx 0.48$ GeV.

C. Confining Force

• The confining force is defined via hydrostatic equilibrium as $f_{\text{conf}}(r) = -2s(r)/r$.
• In the Canonical gauge, $f_{\text{conf}}$ diverges at $r \to 0$ due to the singular shear force. While mathematically consistent with the pressure gradient, this cancellation is physically dubious.
• In the Belinfante gauge, the force is finite and peaks around $r \sim 0.3$ fm.
• The region of negative pressure (often associated with confinement in bag models) extends to larger radii ($r \gtrsim 0.7$ fm) than the peak of the confining force, suggesting the two concepts are not directly equivalent.

D. Equation of State (EoS)

• By eliminating the radial coordinate $r$ between $p(r)$ and $\varepsilon(r)$, the authors construct an EoS for single-nucleon matter.
• The Belinfante-based EoS aligns well with empirical neutron star matter EoS (e.g., SLy4) at high densities.
• The Canonical-based EoS deviates significantly.
• Furthermore, due to strong shear forces inside the nucleon, the isotropic pressure $p(r)$ and radial pressure $p_r(r)$ are inequivalent. This introduces an additional ambiguity: even within a single gauge, it is unclear which pressure definition corresponds to the thermodynamic EoS.

5. Significance and Implications

• Caution in Interpretation: The results suggest that local mechanical properties (energy density, pressure, shear, confining force) extracted from EMT form factors do not have a unique, overarching mechanical significance without specifying the pseudo-gauge.
• Model Dependence: The ambiguity becomes pronounced only when vector fields (spin currents) are included. Models without vector mesons might misleadingly suggest gauge invariance of local distributions.
• Physical Preference: While the Belinfante EMT yields more "reasonable" (finite, non-singular) results for pressure and force distributions, the authors note there is no a priori principle that uniquely selects one gauge over the other. The Canonical EMT remains essential for spin physics (e.g., Jaffe-Manohar decomposition).
• Future Directions: The paper calls for the development of theoretical or phenomenological principles to select the "correct" EMT form or to identify pseudo-gauge-invariant characterizations of nucleon structure. It highlights the need for caution when mapping GPD/DVCS data to local nucleon structures.

In conclusion, the paper establishes that the "internal structure" of the nucleon is not an absolute observable in terms of local mechanical distributions but is contingent on the definition of the Energy-Momentum Tensor, a nuance that must be addressed in future high-precision QCD studies.