On the definition of the nucleon axial charge density

This paper derives and interprets the spatial density distributions of the axial-vector charge density for spin-1/2 systems using sharply localized wave packets in arbitrary Lorentz frames, while also examining the static approximation and its relation to Breit-frame distributions.

The Gist

Imagine you are trying to take a photograph of a tiny, spinning top (a proton or neutron) to see exactly where its "spin charge" is located inside. In physics, this "spin charge" is called the axial charge.

For decades, physicists have had a standard way of taking this picture. They used a specific camera setting (called the "Breit frame") that they thought gave the truest image. However, recent studies suggested that for very small, fast-moving particles, this standard photo might be blurry or even show nothing at all (zero charge), which doesn't make sense because we know these particles do have spin.

This paper is like a team of physicists saying, "Wait a minute, let's try a different camera angle and a different way of focusing." They propose a new method to map out where the axial charge actually lives inside a particle, especially for light particles like protons.

Here is the breakdown of their journey, using some everyday analogies:

1. The Problem: The "Frozen" Photo vs. The "Blurry" Reality

Think of a proton as a buzzing bee.

• The Old Way (Breit Frame): Imagine trying to take a photo of a bee by freezing it in mid-air. You assume the bee isn't moving relative to the camera. In this "frozen" state, the math says the bee's spin charge disappears completely. It's like taking a photo of a spinning fan and saying, "There is no air moving here." It feels wrong.
• The Issue: This "frozen" method works great for heavy, slow objects (like a bowling ball), but it fails for light, fast objects (like the bee) because their quantum nature (their wave-like behavior) gets in the way.

2. The New Solution: The "Sharp Focus" Wave Packet

The authors suggest a new way to define the picture. Instead of freezing the particle, they imagine the particle is inside a wave packet.

• The Analogy: Think of the particle not as a hard marble, but as a cloud of fog. To see the charge inside, you need to focus your camera on a very specific, tiny drop of that fog.
• The "Sharply Localized" Trick: They use a mathematical technique to make this "drop of fog" incredibly small and sharp. By doing this, they can look at the particle in its own "rest frame" (where it isn't moving on average) without forcing it to be frozen in a way that breaks the laws of physics.

3. The Discovery: The Charge Hides in the Spin

When they took this new "sharp focus" picture in the rest frame, they found something surprising: The raw image still looked like it had zero charge.

Why?

• The Metaphor: Imagine the axial charge is a secret message written on a spinning top. If you look at the top from the side while it's spinning, the message blurs into a circle and looks like nothing is there. The charge isn't gone; it's just hidden inside the spin.
• The math showed that the charge density was proportional to the particle's spin (represented by a Pauli matrix, $\sigma$). Because the particle is spinning, the charge "wobbles" so fast in the rest frame that it averages out to zero unless you account for the spin.

4. The Fix: Stripping Away the Spin

The authors realized that to get the true "map" of the charge, you have to remove the spin factor from the equation.

• The Analogy: It's like realizing the "blur" you saw earlier was just the motion of the camera. If you mathematically "stabilize" the camera (remove the spin-dependent factor), the hidden message appears clearly.
• Once they did this, they got a clean, 3D map of the axial charge density.

5. The Results: Different Angles, Different Views

The paper also looked at what happens if you view the particle from different speeds:

• The "Slow" View (Zero Average Momentum Frame): This gives a 3D map of the charge. It's the most accurate for light particles like protons.
• The "Fast" View (Infinite Momentum Frame): If you zoom past the particle at near light speed, the 3D map squishes flat, becoming a 2D pancake (a disk). This is similar to how a spinning coin looks like a line when it's spinning fast.
• The "Heavy" View (Static Limit): If the particle is very heavy (like a bowling ball), the new method agrees with the old "Breit frame" method. The old way wasn't wrong, just limited to heavy objects.

Why Does This Matter?

This is important because it fixes a long-standing confusion in nuclear physics.

1. It works for everyone: The new definition works for both heavy particles (where the old way worked) and light particles (where the old way failed).
2. It clarifies the "Axial Radius": Physicists measure the size of a proton's spin charge (the axial radius). The old method gave one number, but the new, more accurate method suggests a slightly different size. This helps us understand the internal structure of matter more precisely.
3. It resolves the "Vanishing Charge" mystery: It explains why previous calculations showed the charge disappearing—it wasn't disappearing; it was just being masked by the particle's spin in the wrong frame of reference.

In summary: The authors built a better "camera" for the quantum world. They showed that to see the true shape of a particle's spin charge, you can't just freeze the particle; you have to account for its quantum "wobble" and spin. Once you do that, the charge is there, waiting to be mapped.

Technical Summary

1. Problem Statement

The paper addresses the long-standing ambiguity in defining and interpreting the three-dimensional spatial density distributions of local operators, specifically the axial-vector charge density, for spin-1/2 systems (such as nucleons).

• Traditional Approach: Historically, the spatial density has been defined as the three-dimensional Fourier transform of the form factor in the Breit frame (where the energy transfer is zero). This approach assumes a static approximation.
• The Conflict: Recent studies (e.g., Ref. [26]) have demonstrated that for spin-1/2 systems, the axial charge density vanishes identically in the Breit frame. Consequently, the Breit-frame approach fails to yield a non-trivial axial charge radius or a meaningful spatial distribution for light systems (where the system's size is comparable to its Compton wavelength).
• The Gap: There is a need for a rigorous definition of axial charge density that:
  1. Is valid for systems of any mass (including light hadrons).
  2. Possesses a proper probabilistic interpretation.
  3. Does not rely on the static approximation which breaks down for relativistic systems.

2. Methodology

The authors employ a framework based on sharply localized wave packet states rather than momentum eigenstates. This approach, developed in their previous works [20, 22–25], allows for a proper definition of spatial densities in specific reference frames.

• Wave Packet Construction: They define a state $|\Phi, X, s\rangle$ using a spherically symmetric, sharply localized wave packet $\phi(p)$ with a localization parameter $R$. The limit $R \to 0$ corresponds to a sharply localized state.
• Reference Frames:
  • Zero Average Momentum Frame (ZAMF): The frame where the mean momentum of the wave packet is zero.
  • Moving Frames: General frames with velocity $v$.
  • Infinite Momentum Frame (IMF): The limit where $v \to 1$.
• Calculation Technique:
  • They calculate the matrix element of the axial charge density operator $j^0_A$ between wave packet states.
  • They utilize the method of dimensional counting (and the "strategy of regions") to evaluate the $R \to 0$ limit of the integrals without specifying the exact form of the form factors or profile functions, provided they fall off sufficiently fast at high momentum transfer.
  • They compare two definitions:
    1. Exact ZAMF/IMF: No static approximation.
    2. "Naive" Static Limit: Expanding in powers of $1/m$ (static approximation) before taking the localization limit.

3. Key Results

A. Vanishing Densities in ZAMF

When calculating the spatial density $J^0_{A, \text{ZAMF}}(r)$ using sharply localized wave packets in the ZAMF without static approximation, the result is identically zero.

• Reason: The resulting expression is proportional to the Pauli spin vector $\vec{\sigma}$ dotted with a vector derived from the integration. Due to the spherical symmetry of the wave packet in the ZAMF, there is no preferred direction to contract with $\vec{\sigma}$ to form a rotationally invariant scalar density. Thus, the integral vanishes.
• Implication: This confirms the findings of Ref. [26] that the naive Breit-frame interpretation (which is a static limit) is insufficient for defining the axial charge density of spin-1/2 systems.

B. Moving Frames and the Spin-Dependent Factor

In moving frames (velocity $v$), the density $J^0_{A, v}(r)$ is non-zero but exhibits a specific structure:
$$J^0_{A, v}(r) = (\vec{v} \cdot \vec{\sigma}) \rho_{A, v}(r)$$
The density is a pseudoscalar (it changes sign under spatial reflection) because it is proportional to the spin vector $\vec{\sigma}$. This behavior is unphysical for a "charge density," which should be a scalar quantity.

C. Proposed Definition of Axial Charge Density

The authors argue that the physical axial charge density $\rho_A$ should be defined by factoring out the kinematic, spin-dependent prefactor $(\vec{v} \cdot \vec{\sigma})$ which encodes the frame dependence and spin orientation but not the internal structure.

By defining $\rho_{A, v}(r)$ such that $J^0_{A, v}(r) = (\vec{v} \cdot \vec{\sigma}) \rho_{A, v}(r)$, they obtain frame-independent (or frame-covariant) densities that represent the true internal structure:

1. ZAMF Definition (Relativistic, no static limit):
   $$\rho_{A, \text{ZAMF}}(r) = \frac{1}{4\pi} \int d\hat{m} \int \frac{d^3q}{(2\pi)^3} G_A\left[(\hat{m} \cdot q)^2 - q^2\right] e^{-i q \cdot r}$$
   This definition is valid for systems where the Compton wavelength is comparable to the system size.

2. IMF Definition:
   $$\rho_{A, \text{IMF}}(r) = \delta(r_\parallel) \int \frac{d^2q_\perp}{(2\pi)^2} G_A(-q_\perp^2) e^{-i q_\perp \cdot r_\perp}$$
   This reduces to the well-known 2D transverse distribution multiplied by a longitudinal delta function.

3. Naive/Breit-Frame Definition (Static Limit):
   If one takes the static limit ($1/m$ expansion) first, the density reduces to the traditional Breit-frame Fourier transform:
   $$\rho_{A, \text{naive}}(r) = \int \frac{d^3q}{(2\pi)^3} G_A(-q^2) e^{-i q \cdot r}$$
   This is appropriate only for heavy systems ($m \to \infty$).

D. Axial Charge Radius Discrepancy

The paper highlights a crucial difference in the definition of the mean-square axial radius $\langle r^2 \rangle_A$:

• Relativistic (ZAMF) Definition: $\langle r^2 \rangle_A = 4 G'_A(0)$
• Naive (Static/Breit) Definition: $\langle r^2 \rangle_{A, \text{naive}} = 6 G'_A(0)$

The factor of $6/4 = 1.5$ difference arises because the static approximation does not commute with the sharp localization limit ($R \to 0$). For light hadrons (like the nucleon), the relativistic definition is the physically correct one.

4. Significance and Contributions

• Resolution of the "Vanishing" Paradox: The paper explains why the axial charge density vanishes in the Breit frame/ZAMF (due to the pseudoscalar nature of the operator in a symmetric frame) and provides a mathematically consistent way to extract a non-zero, physically meaningful density.
• Generalization to Light Systems: Unlike the Breit-frame approach, the new ZAMF definition is applicable to light systems where relativistic effects are dominant and the Compton wavelength is not negligible.
• Clarification of "Naive" vs. "Proper" Definitions: It rigorously distinguishes between the static approximation (valid for heavy systems) and the fully relativistic treatment required for nucleons.
• Correction of Axial Radius: It suggests that the conventional extraction of the axial radius from the slope of the form factor ($6 G'_A(0)$) may need reinterpretation or correction ($4 G'_A(0)$) when defining the physical spatial distribution of the nucleon, depending on the theoretical framework used.
• Universality: The derived densities for the axial charge coincide with the electric charge densities (from Ref. [20]) upon replacing the electric form factor with the axial form factor, unifying the treatment of different current types.

In summary, the authors provide a robust, relativistic definition of the nucleon axial charge density using sharply localized wave packets, resolving previous contradictions and establishing a framework applicable to both heavy and light hadronic systems.