⚛️ nuclear theory

Chiral anomaly and nucleon spin decomposition in the chiral quark soliton model

This study extends the chiral quark soliton model to include a flavor-singlet pseudoscalar field generated by the chiral anomaly, revealing that while the anomaly-induced field enhances quark helicity and its associated mass suppresses this effect in favor of orbital angular momentum, these modifications result in only modest redistributions that do not qualitatively alter the conventional nucleon spin decomposition.

Original authors: Josuke Minamiguchi, Tomoya Uji

Published 2026-09-25

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CC BY 4.0

Original authors: Josuke Minamiguchi, Tomoya Uji

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 proton and neutron, the building blocks of the visible universe, a fundamental mystery has long puzzled physicists: where does the particle's spin come from? Spin is an intrinsic form of angular momentum, a quantum property that makes these particles behave like tiny, spinning tops. For decades, scientists assumed that the spin of a proton was simply the sum of the spins of the three quarks that make it up. However, experiments revealed a startling truth: the quarks themselves contribute far less than expected. The missing spin must be hiding somewhere else, perhaps in the swirling orbital motion of the quarks or in the gluons that bind them together. Understanding how this spin is distributed is crucial for a complete picture of matter, and it requires peering into the complex, invisible forces that govern the subatomic world.

A team of researchers at the University of Tokyo has taken a fresh look at this puzzle by refining a theoretical model that describes how quarks move inside a nucleon. They focused on a specific, subtle effect known as the chiral anomaly. In the quantum world, certain symmetries that should exist in theory are broken in practice, and this breaking generates a unique force that influences how particles behave. The researchers wanted to see how this anomaly, which is responsible for giving mass to a specific type of particle called the eta-prime meson, changes the balance between the quarks' own spin and their orbital motion. By adjusting the strength of this anomaly in their calculations, they could observe how the internal dynamics of the nucleon shift in response.

The scientists used a framework called the chiral quark soliton model, which treats the nucleon as a stable, self-consistent field of quarks moving within a background of meson fields. In their standard version, this model only includes the pion, a light particle that mediates the strong force between quarks. To investigate the anomaly, the team expanded the model to explicitly include a heavier, flavor-singlet pseudoscalar field. This field represents a mode of vibration that would be massless if not for the chiral anomaly, which gives it a significant mass. The researchers treated this mass as a variable knob they could turn. By keeping all other parameters fixed and only changing the mass of this singlet field, they could isolate exactly how the anomaly influences the distribution of spin.

In their simulations, the researchers calculated the probability of finding a quark with its spin aligned with the nucleon's overall spin, as well as the contribution from the quarks' orbital motion. They found that the presence of this singlet field, induced by the rotation of the nucleon, enhances the contribution of the quark spin. When the anomaly-induced mass of the singlet field was set to a lower value, the quark spin contribution increased, while the orbital motion contribution decreased. Conversely, when they increased the mass of this field, the quark spin contribution dropped, and the orbital motion took up more of the slack. This suggests that the chiral anomaly acts as a regulator, suppressing the quark spin contribution when the anomaly is strong.

Despite these shifts, the total amount of spin remained constant, satisfying the fundamental conservation laws of physics. The researchers observed that changing the mass of the singlet field from zero up to a value of 696 MeV caused the quark spin contribution to vary from about 0.407 down to 0.379. While this is a measurable change, the overall redistribution of spin was modest. The study indicates that while the chiral anomaly does influence how spin is shared between the quarks' intrinsic spin and their orbital motion, it does not fundamentally alter the picture provided by the conventional model. The anomaly-induced field adds a layer of complexity, enhancing the quark spin contribution relative to a model without it, but the effect is not large enough to qualitatively change our understanding of the nucleon's spin structure.

The findings offer a clearer, more nuanced view of the proton's interior. The researchers demonstrated that the chiral anomaly, through its effect on the singlet field, creates a specific response in the quark sea that favors the alignment of quark spins. However, this effect is counterbalanced by the inertia of the system, leading to a delicate equilibrium. The study confirms that the spin of the nucleon is a shared responsibility, with the quarks' spin and their orbital motion constantly adjusting to one another based on the underlying quantum forces. While the chiral anomaly plays a role in this balance, it is just one part of a much larger, intricate dance of forces that holds the universe together. The work provides a solid theoretical foundation for future experiments, such as those planned at the Electron-Ion Collider, which aim to measure these distributions with unprecedented precision.

Technical Summary: Chiral Anomaly and Nucleon Spin Decomposition in the Chiral Quark Soliton Model

Problem Statement
The paper addresses the theoretical challenge of understanding how the chiral anomaly, specifically the $U(1)_A$ anomaly, influences the decomposition of the nucleon spin between quark helicity and orbital angular momentum (OAM). While the flavor-singlet sector of the nucleon spin is known to be connected to gluon topology via the chiral anomaly, the quantitative impact of flavor-singlet pseudoscalar dynamics on the size of the quark helicity contribution remains an open question. Standard two-flavor Chiral Quark Soliton Models (CQSM) typically retain only the pion field, effectively freezing out the flavor-singlet pseudoscalar mode due to its large anomaly-induced mass. This work seeks to explicitly investigate how the response of this singlet mode in a polarized nucleon affects the partition of spin between quark helicity and OAM.

Methodology
The authors extend the conventional two-flavor CQSM by explicitly retaining the flavor-singlet pseudoscalar field, $\eta_0$ , and its anomaly-induced mass, $m_0$ .

Theoretical Framework: The model is formulated within the large- $N_c$ limit and spontaneous chiral symmetry breaking. The effective action includes constituent quarks coupled to a $U(2)$ chiral field, $U(x) = e^{i\eta_0(x)/f_0}U_{\pi}(x)$ , and a Witten–Di Vecchia–Veneziano potential term to account for the anomalous breaking of $U(1)_A$ symmetry.
Collective Quantization: Nucleon states are constructed by quantizing the collective rotational zero modes of the static hedgehog soliton. While the static hedgehog has $\eta_0 = 0$ , the collective rotation induces a non-zero $\eta_0$ response field, parametrized as $\eta_0(r, t) \propto \Omega_a(t)\hat{r}_a$ , where $\Omega$ is the angular velocity.
Calculations: The authors derive expressions for the quark helicity distribution $[\Delta u + \Delta d](x)$ $[Δ u + Δ d] (x)$ and the orbital angular momentum distribution $q_L(x)$ $q_{L} (x)$ to first order in the collective angular velocity. These distributions include contributions from:
1. The conventional rotation of the pion field ( $\Omega$ contribution).
2. The explicit insertion of the rotation-induced singlet field ( $\eta$ contribution).
3. The normalization of the collective rotation, which depends on the total moment of inertia $I = I_{\Omega\Omega} + I_{\eta}$ .
Numerical Implementation: The model is solved self-consistently using a Pauli–Villars regularization scheme. The anomaly-induced mass $m_0$ is treated as a tunable parameter, varied from $0$ to $696$ MeV (and the limit $m_0 \to \infty$ ), while other parameters (constituent quark mass $M$ , pion decay constant $f_\pi$ ) are fixed to physical values.

Key Contributions

Extension of CQSM: The paper provides a consistent field-theoretical formulation of the CQSM that explicitly includes the flavor-singlet pseudoscalar degree of freedom and its anomaly-induced mass, allowing for the calculation of singlet-induced responses in polarized nucleons.
Decomposition of Spin Observables: The authors derive and calculate the specific contributions of the induced singlet field to the quark helicity and OAM distributions, separating them from the conventional pion-only contributions.
Parameter Dependence: The study systematically examines the sensitivity of the nucleon spin decomposition to the strength of the $U(1)_A$ anomaly by varying $m_0$ .

Results

Induced Field Profile: The rotation-induced $\eta_0$ profile is non-zero and its magnitude is inversely related to $m_0$ . Lowering $m_0$ enhances the peak of the induced field and broadens its spatial profile.
Spin Redistribution:
- Increasing the anomaly-induced mass $m_0$ suppresses the quark helicity contribution ( $\Delta\Sigma$ ) and increases the orbital angular momentum contribution ( $L_q$ ).
- Conversely, decreasing $m_0$ (weakening the anomaly) enhances the quark helicity contribution.
- Specifically, as $m_0$ decreases from infinity to $0$, the total quark helicity fraction ( $\frac{1}{2}\Delta\Sigma$ ) rises from $0.186$ to $0.203$, while $L_q$ falls from $0.315$ to $0.299$.
Magnitude of Effect: Despite the trend, the redistribution is quantitatively modest. At the physical value $m_0 \approx 696$ MeV, the singlet contribution accounts for only about $4\%$ of the total quark helicity and $1.5\%$ of the OAM. The difference between the full model and the conventional pion-only model ( $m_0 \to \infty$ ) is small (e.g., $\Delta\Sigma$ changes from $0.373$ to $0.379$).
Spin Sum Rule: The calculated distributions satisfy the nucleon spin sum rule ( $\frac{1}{2}\Delta\Sigma + L_q = 1/2$ ) to better than $1\%$ across all tested values of $m_0$ .

Significance and Claims
The paper claims that while the chiral anomaly and the associated flavor-singlet pseudoscalar dynamics do influence the partition of nucleon spin, this influence is modest within the parameter space considered.

The induced singlet field enhances the quark helicity contribution relative to the conventional pion-only model, but increasing the anomaly-induced mass suppresses this enhancement.
The authors conclude that the inclusion of the singlet field does not qualitatively alter the spin decomposition established by the conventional CQSM. The redistribution of spin between helicity and OAM remains small, suggesting that the qualitative picture of nucleon spin in the CQSM is robust against variations in the anomaly-induced mass.
The work highlights the competing roles of the singlet field and the topological susceptibility in determining the magnitude of the helicity variation, offering a spatial interpretation where the helicity enhancement at lower $m_0$ corresponds to a strengthening of the topological dipole structure.

The authors suggest that a natural extension of this work would be to calculate flavor-singlet polarized Generalized Parton Distributions (GPDs) to investigate these dynamics at finite momentum transfer.

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