Energy-momentum tensor form factor D(t) of proton and neutron

This paper constructs a neutron model analogous to a classical proton model to demonstrate that, despite electromagnetic-induced divergences at extremely small momentum transfers, the energy-momentum tensor form factor $D(t)$ of protons and neutrons remains practically indistinguishable across experimentally accessible ranges, thereby accurately explaining the nucleon mass difference and reproducing lattice QCD data.

The Gist

Imagine the proton and the neutron not as tiny, solid marbles, but as bustling, invisible cities made of "dust" (quarks and gluons) held together by invisible forces. Inside these cities, there is a constant tug-of-war: forces pushing the dust apart and forces pulling it together. Physicists call the map of these internal forces the Energy-Momentum Tensor (EMT), and a specific number on that map, called the D-term, tells us how "stiff" or "squishy" the particle is.

This paper, written by Andrea Mejia and Peter Schweitzer, asks a fascinating question: If we look at the internal pressure map of a proton and a neutron, do they look different?

Here is the story of their findings, explained through simple analogies.

1. The Two Cities: Proton vs. Neutron

Think of the Neutron as a quiet, neutral city. It has no electric charge. The only forces at play are the "Strong Nuclear Forces" (like super-strong glue) that hold the city together. Because there is no electric repulsion, the city is compact and tidy. Its internal pressure map (the D-term) is a smooth, negative curve that stays stable no matter how closely you look at it.

Now, think of the Proton as a city that is slightly different: it has a positive electric charge. This is like having a crowd of people in the city who are all holding balloons that repel each other.

• The Problem: Because the proton is charged, the "balloons" (electromagnetic forces) push the dust apart.
• The Mathematical Glitch: In the world of pure math, this repulsion creates a weird problem. If you try to calculate the D-term for the proton at extremely tiny distances (or very low energy), the number goes crazy—it shoots up to infinity. It's like trying to measure the pressure of a balloon that keeps expanding forever; the math breaks.

2. The "Regularized" Solution

The authors realized that while the math says the proton's D-term should be infinite, nature doesn't actually work that way in the real world. The "infinity" is an artifact of looking at the problem with a specific mathematical lens that includes the long-range reach of electricity.

To fix this, they invented a concept called "Regularization."

• The Analogy: Imagine you are trying to measure the weight of a backpack, but the backpack has a tiny, invisible feather attached to it that keeps getting heavier the closer you get to it. To get a useful number, you decide to "clip off" the feather and measure just the backpack.
• The Result: When they "clipped off" the infinite electromagnetic part of the proton, the remaining number (the "Regularized D-term") was almost identical to the neutron's number.

3. The Big Discovery: They Look the Same

The paper's main conclusion is surprising and comforting: For all practical purposes, the proton and the neutron are twins.

• The Scale of Difference: The authors calculated that the proton and neutron D-terms look exactly the same down to a scale of about $10^{-4}$ GeV².
• The "Magic" Distance: The proton only starts to look different (and its D-term starts to change sign and diverge) at a distance so incredibly small ($10^{-5}$ to $10^{-8}$ GeV²) that it is far, far smaller than anything we can currently measure with our most powerful microscopes (like the Large Hadron Collider or the future Electron-Ion Collider).

The Metaphor: Imagine two identical twins standing 100 feet away from you. They look exactly the same. You need a telescope to see them clearly. But to see the tiny difference in their eye color (the electromagnetic effect), you would need a telescope powerful enough to see a single atom on their eyelashes. Our current telescopes aren't strong enough yet.

4. Why Does This Matter?

This is a big deal for physicists studying "Hard Exclusive Reactions" (high-energy collisions where we try to see inside particles).

• Before this paper: Scientists might have worried, "Do I need to use a totally different math formula for protons because they are charged, and a different one for neutrons?"
• After this paper: The answer is No. Because the difference is so tiny and happens at a scale we can't reach, physicists can treat the proton and neutron as having the same internal "stiffness" (D-term) in almost all experiments. They can use the simpler, cleaner "neutron" math for the proton, too.

5. The Mass Difference Mystery

The paper also solved a small side-mystery: Why is a proton slightly heavier than a neutron?

• The Cause: It's the "balloon" effect again. The electric repulsion inside the proton pushes the dust apart, requiring more energy to hold the city together.
• The Calculation: Their model predicted this mass difference perfectly (about 0.95 MeV), matching the most advanced computer simulations (Lattice QCD). This proves their model of the "dust city" is a good representation of reality.

Summary

In simple terms, this paper tells us that protons and neutrons are practically identical in their internal mechanical structure. The fact that the proton is charged does create a mathematical "infinity" at the very smallest scales, but that scale is so microscopic that it doesn't matter for any experiment we can do today or in the foreseeable future.

So, for all practical purposes, the proton and the neutron are the same. The "infinity" is just a mathematical ghost that we don't need to worry about unless we build a machine a billion times more powerful than anything we have now.

Technical Summary

1. Problem Statement

The Energy-Momentum Tensor (EMT) form factor $D(t)$, often called the D-term, encodes crucial information about the internal mechanical structure (pressure and shear forces) of hadrons.

• Theoretical Context: In systems governed solely by strong (short-range) forces, $D(t)$ is finite and negative. However, for charged particles, the inclusion of electromagnetic (long-range) forces causes $D(t)$ to change sign at small momentum transfer $t$ and diverge as $1/\sqrt{-t}$ as $t \to 0$. This is a universal QED effect.
• The Core Question: Can this QED-induced divergence be observed experimentally for the proton? Furthermore, must protons and neutrons be treated as fundamentally different objects in phenomenological studies of hard exclusive reactions, or are their mechanical properties effectively indistinguishable at experimentally accessible scales?
• Gap: While the divergence is theoretically established for the proton, a direct comparison with the neutron (which lacks the divergence) using a unified, realistic framework that incorporates both QED and QCD constraints was lacking.

2. Methodology

The authors employ a classical mean-field model (based on the Bia lynicki-Birula model) to describe the nucleon as a system of "dust" bound by scalar ($\sigma$) and vector ($\omega$) meson fields, with the proton additionally subject to the electromagnetic field.

• Model Construction:
  • Proton Model: Uses the established classical model where dust particles interact via attractive scalar forces, repulsive vector forces, and Coulomb repulsion.
  • Neutron Model: Constructed analogously by taking the limit of the electromagnetic coupling $e \to 0$. This renders the dust electrically neutral, eliminating the Coulomb term while retaining the strong interaction dynamics.
• Regularization: Since the proton's $D(t)$ diverges due to the long-range Coulomb tail, the authors utilize and generalize a "regularized" D-term ($D_{prot,reg}$) proposed in previous work. This involves subtracting the divergent Coulomb contribution analytically to yield a finite value comparable to the neutron's $D(t)$.
• Rescaling for Realism: The original model parameters (derived from nuclear matter) underestimate the proton charge radius and the magnitude of the D-term compared to Lattice QCD. To address this, the authors introduce two scaling parameters:
  • $\lambda_m$: Rescales meson masses to correct the proton charge radius to the experimental value ($0.83$ fm).
  • $\lambda_g$: Rescales strong coupling constants to match the magnitude of the D-term found in Lattice QCD.
• Validation: The rescaled model is validated against:
  1. Lattice QCD data for the nucleon $D(t)$ up to $(-t) \approx 1 \text{ GeV}^2$.
  2. Analytical QED predictions for the divergence at $t \to 0$.
  3. The electromagnetic proton-neutron mass difference.

3. Key Contributions

• Unified Neutron-Proton Framework: The construction of a consistent classical neutron model that isolates electromagnetic effects, allowing for a direct comparison of mechanical properties between the two nucleons.
• Quantitative Mass Difference: The model successfully reproduces the electromagnetic contribution to the proton-neutron mass difference ($\approx 0.95$ MeV), aligning with Lattice QCD+QED results.
• Generalized Regularization: The authors provide an independent derivation and generalization of the regularization method for the proton D-term, proving its physical consistency by showing it yields a value nearly identical to the neutron's finite D-term.
• Definition of Neutron Size: The paper argues that the neutron's "size" cannot be defined by its mean square charge radius (which is negative) but is well-defined via EMT radii (mechanical and mass radii).

4. Key Results

• Indistinguishability at Experimental Scales:
  • For momentum transfers $(-t) \gtrsim 10^{-4} \text{ GeV}^2$, the $D(t)$ form factors of the proton and neutron are practically indistinguishable.
  • The proton's $D(t)$ only begins to deviate significantly from the neutron's at $(-t) \lesssim 10^{-5} \text{ GeV}^2$, where it changes sign and eventually diverges.
  • Current and planned experiments (e.g., Jefferson Lab, Electron-Ion Collider) operate at $(-t)_{min} \approx 0.03 - 0.09 \text{ GeV}^2$, which is orders of magnitude too large to observe the QED divergence.
• Lattice QCD Agreement: After rescaling ($\lambda_m = 0.85, \lambda_g \approx 3.2$), the model reproduces Lattice QCD data for the nucleon $D(t)$ with high precision ($\chi^2_{d.o.f.} \approx 1.05$) up to $(-t) \approx 1 \text{ GeV}^2$.
• EMT Radii:
  • The neutron is systematically slightly smaller than the proton across all defined radii (dust, mechanical, and mass).
  • Mechanical Radius: $\langle r^2_{mech} \rangle^{1/2}_{neut} \approx 0.769$ fm vs. $\approx 0.797$ fm (regularized proton).
  • Mass Radius: $\langle r^2_{mass} \rangle^{1/2}_{neut} \approx 0.755$ fm vs. $\approx 0.759$ fm (regularized proton).
  • The slight "swelling" of the proton is attributed to Coulomb repulsion.
• QED Divergence Confirmation: The model analytically recovers the universal QED divergence $D(t) \sim \frac{\alpha \pi M}{4\sqrt{-t}}$ as $t \to 0$, confirming the theoretical prediction.

5. Significance and Conclusions

• Phenomenological Implication: For all practical purposes in current and foreseeable future phenomenological studies of hard exclusive reactions, the proton and neutron D-form factors should be treated as identical. The QED divergence is an asymptotic effect inaccessible to experiment.
• Regularized D-term: The concept of a "regularized proton D-term" is validated as a robust physical quantity. It allows phenomenologists to use a finite, negative D-term for the proton (similar to the neutron) in the experimentally accessible region $(-t) > 10^{-4} \text{ GeV}^2$.
• Neutron Characterization: The study establishes that EMT radii provide a meaningful definition of the "size" of the neutron, resolving the ambiguity caused by the negative charge radius.
• Theoretical Bridge: The work successfully bridges the gap between low-energy effective models, Lattice QCD (strong force regime), and QED (long-range regime), demonstrating that a simple classical mean-field approach can capture complex nucleon structure properties when properly scaled.

In summary, the paper concludes that while the proton and neutron are theoretically distinct due to electromagnetic effects, these differences are negligible at experimentally accessible energy scales. Consequently, the mechanical structure of the nucleon can be effectively described by a single, universal D-term profile for both particles in phenomenological applications.