Quantum stress and torsion distributions in the deuteron

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Imagine the deuteron (the nucleus of heavy hydrogen, made of one proton and one neutron) not as a tiny, static ball, but as a bustling, microscopic city. Inside this city, particles are zooming around, pushing, pulling, and spinning.

For a long time, physicists have been trying to map out the "streets" and "buildings" of this city. But recently, they've started asking a different question: What does the weather feel like inside? Is it windy? Is there pressure? Are the buildings being squeezed or stretched?

This paper by Wim Cosyn, Adam Freese, and Alan Sosa is like a new, ultra-high-resolution weather map for the deuteron. They didn't just look at where the particles are; they calculated the forces, pressures, and stresses holding this tiny city together.

Here is a breakdown of their findings using simple analogies:

1. The "Stress Map" (The Energy-Momentum Tensor)

In physics, there's a mathematical object called the "Energy-Momentum Tensor." Think of this as the blueprint of all the forces inside the deuteron.

The Old Way: Previous studies only looked at the "symmetric" part of this blueprint. Imagine looking at a building and only measuring how much it's being squished from the top and bottom.
The New Way: This paper calculates all 11 parts of the blueprint, including the "asymmetric" parts. This is like realizing the building is also being twisted and sheared sideways. They found that the deuteron isn't just being squeezed; it's being twisted by the spinning of its internal parts.

2. The "Twist" (Torsion Stress)

One of the most exciting discoveries is something called torsion stress.

The Analogy: Imagine holding a wet towel and twisting it. The fibers inside are being reoriented.
In the Deuteron: The proton and neutron have "spin" (like tiny tops). As they move between different orbital shapes (called S-waves and D-waves), the forces between them act like a wrench, twisting the spin of the quarks inside. This paper maps out exactly where this "twisting" happens. It's like finding the exact spot on a rubber band where the tension is highest just before it snaps.

3. The "Pressure Cooker" (Radial vs. Tangential Pressure)

If you look at the pressure inside the deuteron, it's not the same everywhere.

The Center: The middle of the deuteron is under high pressure, like the center of a star or a deep-sea diver. The particles are pushing outward, trying to expand.
The Edge: As you move toward the outside, the pressure flips. Instead of pushing out, the forces turn into tension (pulling inward), like the skin of a balloon or the surface of a water droplet. This "skin" holds the deuteron together, preventing it from flying apart.
The Shape: Because the deuteron isn't a perfect sphere (it's a bit like a peanut or a dumbbell), the pressure isn't uniform. It's stronger in some directions than others, creating a complex, 3D stress pattern.

4. The "Force Lines" (What the Nucleons Feel)

The authors used a famous physics rule (the Cauchy momentum equation) to figure out what the proton and neutron actually "feel."

The Result: They found that the nucleons are being pushed apart at very short distances (repulsion) and pulled together at medium distances (attraction).
The Surprise: The forces aren't just straight lines pointing in or out. Because of the spin and the "twisting" mentioned earlier, the forces are curved and swirling. It's as if the nucleons are running on a track that is constantly curving, rather than a straight road.

5. Why Does This Matter?

You might ask, "Why do we care about the stress inside a hydrogen nucleus?"

The "Glue" Mystery: We know protons and neutrons stick together, but the "glue" (the strong nuclear force) is incredibly complex. By mapping the stress, we are essentially seeing the mechanical blueprint of the strong force.
Testing Our Theories: The authors compared their map to other maps made by different teams. They found that while some parts agreed, others disagreed. This tells us that our understanding of the deuteron's internal "weather" is still a bit foggy, and we need better tools to clear it up.
The "Non-Conserved" Clue: In their calculation, they found some forces that didn't perfectly balance out (called "non-conserved" form factors). In a closed system, forces should balance. The fact that they don't here is actually a good thing! It proves that the proton and neutron are an "open system" exchanging forces with each other. It's like seeing the wind blowing through a window; it tells you there's a connection to the outside world (the force carriers, like pions).

Summary

This paper is a 3D stress test for the deuteron.

They found that the deuteron is a twisted, spinning, pressurized object.
It has a hot, pressurized core and a tense, skin-like edge.
The forces inside are curved and swirling, not just straight lines.
They discovered a new kind of twisting force (torsion) that reorients the spins of the particles inside.

By understanding these mechanical properties, we get a deeper, more "human" understanding of how the universe holds itself together at the smallest scales. It's like moving from a black-and-white sketch of a building to a full-color, 3D simulation of how the wind, weight, and tension interact within its walls.

1. Problem Statement

The mechanical properties of hadrons, described by the Energy-Momentum Tensor (EMT) form factors (EMT-FFs), have been a subject of intense study in recent years. While significant progress has been made for spin-0 and spin-1/2 systems (like the proton), the deuteron (a spin-1 system) remains under-explored.

Gap in Literature: Previous calculations of deuteron EMT-FFs were limited. Most studies calculated only six of the eleven possible form factors for the asymmetric EMT, often using relativistic reduction techniques or specific models (light-front, chiral EFT, holographic QCD) that introduced ambiguities regarding relativistic density definitions.
The Challenge: A complete description of the deuteron's internal mechanical structure requires calculating all eleven form factors of the asymmetric EMT. This includes "non-conserved" form factors, which are often discarded but are physically significant for describing forces on subcomponents (nucleons) within the impulse approximation.
Objective: To provide a comprehensive calculation of all eleven EMT-FFs for the deuteron using ordinary non-relativistic quantum mechanics, derive the associated spatial distributions (mass, momentum, stress, forces), and interpret the physical meaning of the asymmetric (torsion) components.

2. Methodology

The authors employ a rigorous non-relativistic framework based on the impulse approximation and the Galilei symmetry group.

Framework:
- Non-Relativistic Quantum Mechanics: The calculation avoids the ambiguities of relativistic density definitions by treating the deuteron as a non-relativistic system. This allows for the use of high-precision phenomenological potentials.
- Wave Function: The AV18 deuteron wave function is used, known for its high accuracy in describing nucleon-nucleon interactions.
- Impulse Approximation: The calculation considers only one-body contributions (proton and neutron currents). Consequently, the energy-momentum tensor is not locally conserved, leading to non-zero "non-conserved" form factors.
Formalism:
- EMT Decomposition: The authors introduce a new decomposition of the spin-1 EMT matrix element into eleven form factors:
  - Symmetric (9): $A_U, A_T, J, D_U, D_{T1}, D_{T2}, \bar{c}_U, \bar{c}_{T1}, \bar{c}_{T2}$ .
  - Asymmetric (2): $S, \bar{s}$ .
- Analytic Derivation: They derive analytic formulas for these form factors in terms of the deuteron's S-wave ( $u(r)$ ) and D-wave ( $w(r)$ ) radial functions and the nucleon EMT-FFs.
- Spatial Densities: Using the convolution formalism (Galilei boost), they transform the form factors into 3D spatial densities for mass, momentum, mass flux, and stress tensors.
- Force Calculation: The Cauchy momentum equation ( $\nabla \cdot \mathbf{T} = \mathbf{f}$ ) is applied to the stress tensor to derive the force distributions felt by the nucleons.

3. Key Contributions

First Complete Calculation: This is the first study to calculate all eleven EMT-FFs for the deuteron, including the four non-conserved form factors ( $\bar{c}_U, \bar{c}_{T1}, \bar{c}_{T2}, \bar{s}$ ) and the antisymmetric form factors ( $S, \bar{s}$ ).
Non-Relativistic Precision: By utilizing the AV18 wave function within a strict non-relativistic framework, the authors avoid ambiguities associated with relativistic reductions and provide a benchmark for future lattice QCD and chiral EFT comparisons.
Torsion Stress Discovery: The paper explicitly identifies and quantifies torsion stress arising from the antisymmetric part of the stress tensor. It demonstrates that this stress is responsible for the reorientation of fermion spin (nucleon spin) as the system transitions between S- and D-waves.
Force Mapping: The authors map the internal forces within the deuteron, showing how the inter-nucleon force is distributed over the quarks and gluons of the nucleons.

4. Key Results

A. Form Factors

Conserved Form Factors: The results for $A_U, A_T, J, D_U, D_{T1}, D_{T2}$ $A_{U}, A_{T}, J, D_{U}, D_{T 1}, D_{T 2}$ show mixed agreement with prior works (Freese & Cosyn, He & Zahed, Panteleeva et al.).
- Discrepancies in $A_T$ are attributed to different deuteron quadrupole moments arising from different wave functions.
- Discrepancies in $D_{T1}$ and $D_{T2}$ highlight the sensitivity of these factors to the specific dynamics and the treatment of exchange currents.
Non-Conserved Form Factors: The four non-conserved form factors are non-zero. They are physically meaningful as they quantify the forces acting on the nucleons due to the lack of local conservation in the one-body approximation.
Antisymmetric Form Factors:
- $S(0) \approx 0.37$ : Indicates that intrinsic quark spin carries about 37% of the deuteron's total spin (slightly less than the proton due to the deuteron depolarization factor).
- $\bar{s}$ : Arises solely from spin-dependent forces (tensor and spin-orbit coupling).

B. Spatial Distributions

Mass Density: Reproduces the known "dumbbell" ( $m_j=0$ ) and "donut" ( $m_j=\pm 1$ ) shapes, with blurring due to nucleon finite size.
Stress Tensor:
- Principal Stresses: The symmetric stress tensor is diagonalized to find principal stresses (isoradial and isopolar). The isoradial pressure is strictly positive (expansive), while tangential pressures become negative (tensile) at larger distances, similar to a liquid drop but with a diffuse boundary.
- Torsion Stress: The antisymmetric part creates a torsion stress ( $\tau$ ) that induces a vortical momentum flux. This stress is non-zero only where spin-dependent forces are active (specifically at the boundary of the D-wave probability), facilitating the continuous rotation of nucleon spin on the Bloch sphere.
Mechanical Radius: The calculated mechanical radius is $\sqrt{\langle r^2 \rangle}_{\text{Mech}} \approx 1.61$ fm, which is smaller than the mass radius (2.04 fm). This contrasts with the proton (where mechanical > mass radius) and suggests that including two-body currents (force carriers) is crucial for a complete picture.
Force Distributions:
- Forces are short-range repulsive near the center and attractive further out.
- At intermediate distances (~0.5 fm), forces become predominantly polar, directed toward the equator ( $m_j=0$ ) or poles ( $m_j=\pm 1$ ), aligning with regions of high mass concentration.
- The forces are non-radial due to tensor forces and spin-orbit coupling.

5. Significance

Theoretical Benchmark: This work provides a high-precision non-relativistic benchmark for deuteron mechanical properties, essential for validating lattice QCD results and chiral EFT calculations.
Physical Insight into Spin-1 Systems: It clarifies the role of the asymmetric EMT in spin-1 systems, demonstrating that torsion stress is a real, calculable quantum mechanical effect linked to spin reorientation, distinct from classical shear stress.
Force Mapping: By utilizing the Cauchy momentum equation, the paper successfully maps the "force felt by subcomponents," offering a new perspective on how the strong nuclear force is distributed within a nucleus.
Future Directions: The authors outline that future work must include two-body currents (exchange currents) to restore local momentum conservation (making non-conserved form factors vanish) and to refine the mechanical radius. They also suggest applying this framework to heavy quarkonia and separating quark/gluon contributions.

In summary, this paper establishes a complete, non-relativistic description of the deuteron's mechanical structure, revealing the complex interplay between mass distribution, pressure, and the unique torsion stresses generated by quantum spin dynamics.