⚛️ nuclear theory

Nucleon 3D intrinsic spin structure from the weak-neutral axial-vector form factors

This paper investigates the relativistic 3D weak-neutral axial-vector and spin distributions in nucleons, demonstrating that the conventional axial radius is physically meaningless while defining a genuine 3D spin radius dependent on the induced pseudoscalar form factor, and provides corresponding cross-section formulas to guide future neutrino-scattering experiments.

Original authors: Yi Chen

Published 2026-09-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside every atom, the nucleus is built from protons and neutrons, which are themselves made of even smaller particles called quarks. These protons and neutrons are not solid, featureless balls; they are complex, swirling systems with a fundamental property called "spin." Spin is a type of intrinsic angular momentum, a quantum characteristic that makes these particles behave as if they are constantly rotating, even though they are point-like in many ways. Understanding how this spin is distributed inside a proton is one of the great challenges in modern physics, because it helps explain why matter has mass and how the universe is structured. Scientists study this by firing particles at protons and observing how they bounce off, looking for clues hidden in the way the proton reacts. One of the most powerful tools for this is the weak nuclear force, a fundamental interaction that governs how particles decay and interact, which is distinct from the forces that hold the nucleus together.

A researcher has recently taken a fresh look at how the spin of a proton is arranged in three-dimensional space, using the weak nuclear force as their guide. By analyzing the mathematical descriptions of how protons interact with particles called neutrinos, they have mapped out the internal "spin density" of the proton with new precision. Their work clarifies a long-standing confusion in the field regarding how to measure the size of this spin distribution. They found that a quantity scientists had previously used to estimate the proton's spin radius was actually misleading. Instead, they identified a different, more accurate way to define the size of the proton's spin, revealing that the spin is spread out over a much larger area than previously thought.

For decades, physicists have tried to measure the "axial radius" of a proton, a number intended to describe how far the proton's spin extends from its center. This measurement was traditionally calculated using a specific formula based on how the proton reacts to the weak force. However, the researcher in this study demonstrated that this traditional method is fundamentally flawed for a spinning particle like a proton. They showed that the mathematical object used in that old formula does not actually represent a physical distribution of charge or spin in space. Because of the unique way the weak force interacts with the proton's internal structure, the concept of a simple, single-number radius for the proton's spin does not exist in the way it does for other properties. The study proves that the old method of calculation cannot be interpreted as a real, physical size.

Instead of the old method, the researcher developed a new definition for the proton's spin radius that accounts for the full complexity of relativistic physics. They calculated a new value based on the actual distribution of the spin current inside the proton. This new calculation yields a radius of approximately 2.1054 femtometers. To put this in perspective, a femtometer is one quadrillionth of a meter, a scale so small that it is difficult to imagine, yet it is the standard unit for measuring the size of atomic nuclei. This new radius is a physically meaningful quantity, whereas the value of 0.6510 femtometers derived from the older method is not a valid physical radius. The distinction is substantial, indicating that the proton's spin is not concentrated in a tight core but is distributed over a much wider volume.

The study also provides a roadmap for future experiments to verify these findings. The researcher derived a complete set of equations describing how neutrinos and antineutrinos should scatter off protons in a laboratory setting. These equations include the effects of the newly identified spin distribution and the specific forces involved. By using these new formulas, future experiments at facilities that generate neutrino beams can measure the proton's internal structure with greater accuracy. The goal is to confirm the existence of this new, larger spin radius and to better understand the role of a specific component of the weak force that was previously overlooked in these calculations. This work does not just correct a number; it changes the framework scientists use to visualize the internal landscape of the matter that makes up our world.

Technical Summary: Nucleon 3D Intrinsic Spin Structure from the Weak-Neutral Axial-Vector Form Factors

Problem Statement
The paper addresses the physical interpretation of weak-neutral axial-vector form factors (FFs) in the context of the three-dimensional (3D) internal structure of spin-1/2 hadrons, specifically nucleons. While form factors are momentum-space objects extracted from scattering data, their physical meaning resides in position space. A critical issue identified is the lack of a physically meaningful definition for the 3D root-mean-square (RMS) axial radius for spin-1/2 hadrons. Previous literature has often attempted to define an axial radius based on the slope of the axial form factor $G^Z_A(Q^2)$ at $Q^2=0$ . The author argues that due to the parity-odd nature of the axial charge density, such a definition is physically invalid. Furthermore, the relationship between the relativistic 3D axial-vector four-current, the induced pseudotensor form factor $G^Z_T$ , and the intrinsic spin distribution requires clarification within a relativistic framework.

Methodology
The author employs a quantum phase-space formalism to define relativistic 3D densities in the Breit frame.

Formalism: The study utilizes the hadronic matrix elements of the weak neutral-current (NC) operator, parametrized by six form factors: electric ( $G^Z_E$ ), magnetic ( $G^Z_M$ ), induced scalar ( $G^Z_S$ ), axial ( $G^Z_A$ ), induced pseudoscalar ( $G^Z_P$ ), and induced pseudotensor ( $G^Z_T$ ).
Density Definition: The relativistic 3D axial-vector four-current distributions ( $J^\mu_{5,B}(\mathbf{r})$ ) are defined via a Fourier transform of the matrix elements in the Breit frame ( $\mathbf{P}=0$ ).
Spin Relation: The author utilizes the QCD equation of motion to relate the rank-3 weak-neutral canonical spin tensor operator to the axial-vector current, establishing that the spin distribution $\mathbf{S}_B(\mathbf{r})$ is exactly half the axial-vector current distribution $\mathbf{J}_{5,B}(\mathbf{r})$ .
Scattering Cross Sections: To facilitate future experimental extraction of these form factors, the author derives the full tree-level unpolarized differential cross sections for neutrino-proton and antineutrino-proton elastic scattering in the lab frame, explicitly expressing them in terms of $G^Z_A$ , $G^Z_M$ , $G^Z_E$ , and $G^Z_T$ .

Key Contributions and Results

Reinterpretation of Axial Charge Distribution: The paper demonstrates that the relativistic 3D axial charge distribution $J^0_{5,B}(\mathbf{r})$ in the Breit frame is completely described by the induced pseudotensor form factor $G^Z_T(Q^2)$ , not by the standard axial form factor $G^Z_A(Q^2)$ . Specifically, $J^0_{5,B}(\mathbf{r})$ is proportional to the divergence of the spin density involving $G^Z_T$ .
Non-existence of the Axial Radius: The author proves that the quantity $R_A \equiv \sqrt{-6 \frac{1}{G^Z_A(0)} \frac{dG^Z_A}{dQ^2}|_{Q^2=0}}$ , often cited in literature as the RMS axial radius, cannot be interpreted as a physically meaningful 3D radius. Because the axial charge density $J^0_{5,B}(\mathbf{r})$ is parity-odd, the genuine 3D mean-square axial charge radius $\langle r^2_A \rangle$ does not exist for any spin-1/2 hadron.
Definition of a Physical Spin Radius: In contrast to the axial radius, the paper defines a physically meaningful 3D RMS spin radius, $r_{\text{spin}} = \sqrt{\langle r^2_{\text{spin}} \rangle}$ . This is derived from the relativistic 3D weak-neutral spin distribution and is given by:
$\langle r^2_{\text{spin}} \rangle \equiv R^2_A + \frac{1}{4M^2} \left[ 1 + \frac{2G^Z_P(0)}{G^Z_A(0)} \right]$
This definition unambiguously incorporates the induced pseudoscalar form factor $G^Z_P(0)$ .
Numerical Values: Using existing data for the proton's form factors, the author calculates:
- $R_A \approx 0.6510$ fm (noting this is a parameter, not a physical radius).
- $r_{\text{spin}} \approx 2.1054$ fm.
Experimental Framework: The derived differential cross sections for (anti)neutrino-proton elastic scattering provide a theoretical basis for extracting $G^Z_A(Q^2)$ and $G^Z_T(Q^2)$ using neutrino facilities. These measurements would offer constraints complementary to those from charged lepton scattering, specifically helping to constrain the sum of polarized quark Generalized Parton Distributions (GPDs) $\tilde{H}^q$ and $\tilde{T}^q$ .

Significance
The paper claims to investigate the relativistic 3D weak-neutral axial-vector four-current and spin distributions inside a nucleon for the first time. Its primary significance lies in correcting the physical interpretation of the axial form factor $G^Z_A$ and establishing that the "axial radius" is not a valid physical observable for spin-1/2 hadrons. Instead, it proposes the "spin radius" as the correct, physically meaningful metric for the spatial distribution of intrinsic spin.

The work provides a key motivation for the further determination of the induced pseudoscalar form factor $G^Z_P(Q^2)$ , as it is a necessary component for defining the physical spin radius. Additionally, by providing explicit cross-section formulas for neutrino scattering, the paper aims to enable future experimental programs at facilities such as EIC, EicC, or NICA to unveil the nucleon spin structure from a new perspective, linking weak interaction data with quark GPDs.

Technical Summary: Nucleon 3D Intrinsic Spin Structure from the Weak-Neutral Axial-Vector Form Factors

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