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Quark Saturation in Magnetized Pure Neutron Matter

This paper investigates how magnetic fields influence the onset of quark saturation in pure neutron matter by demonstrating that field-induced spin-flavor polarization and transverse spatial squeezing compete to produce a nonmonotonic dependence of the saturation density on field strength.

Original authors: Toru Kojo

Published 2026-10-28

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CC BY 4.0

Original authors: Toru Kojo

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

Deep inside the cores of neutron stars, where matter is crushed to densities far beyond anything found on Earth, the rules of physics begin to blur. Under normal conditions, matter is made of protons and neutrons, which are themselves tiny bundles of even smaller particles called quarks. Usually, these quarks are locked tightly inside their parent protons and neutrons, never to be seen alone. However, as density increases, scientists suspect that these individual particles might start to overlap, eventually forcing the quarks to break free and form a new, fluid state of matter. This transition is not just a theoretical curiosity; it determines how massive a neutron star can grow before collapsing into a black hole. A critical question in this field is knowing exactly when this transition happens. Specifically, researchers want to know when the "seats" available for quarks inside the protons and neutrons become so full that no more can fit, a point known as quark saturation.

A new study by Toru Kojo investigates how powerful magnetic fields change the timing of this saturation. Neutron stars are known to possess the strongest magnetic fields in the universe, particularly a rare type called magnetars. While the fields on the surface of these stars are immense, the internal fields might be even stronger, potentially reaching levels where the magnetic force begins to reshape the very structure of the protons and neutrons themselves. Kojo's work asks a simple but profound question: if you squeeze a neutron star with a magnetic field as strong as those found in these extreme environments, does the matter inside become saturated with quarks sooner or later than it would without the field? To answer this, the researcher built a detailed computer model of neutral particles made of three quarks, simulating how they behave when subjected to magnetic forces that are roughly a billion times stronger than the strongest magnets we can build on Earth.

The study focuses on a specific type of particle found in neutron stars: a neutral baryon composed of two down quarks and one up quark. In the absence of a magnetic field, these particles are somewhat round and their internal parts move in all directions equally. However, when a strong magnetic field is applied, the physics changes dramatically. The magnetic field acts like a powerful lens, forcing the internal motion of the quarks to align. The particles become flattened, like a pancake, squeezing the quarks into a thin disk perpendicular to the magnetic field. This compression happens because the magnetic field restricts how the quarks can move sideways, effectively reducing their freedom of movement to just one dimension. At the same time, the magnetic field rearranges the internal "flavor" and spin of the quarks. It forces the particles into a specific configuration where two of the quarks point their spins in one direction and the third points the other way, a state that is much more stable under these intense conditions.

Kojo used a model based on the idea that protons and neutrons are made of three constituent quarks, treating them like a small system of three balls connected by springs. By calculating the energy and shape of these systems under different magnetic strengths, the researcher could track how the quarks were distributed inside the particles. The key finding is that the magnetic field creates a tug-of-war between two opposing effects. On one hand, the field organizes the quarks so that they crowd into the lowest possible energy states, which makes them want to fill up the available space faster. On the other hand, the magnetic squeezing of the particle's shape spreads the quarks out over a larger area in the sideways direction, which actually makes it harder for them to bump into each other and fill up the space.

The result of this competition is a surprising, non-linear relationship. When the magnetic field is weak, the organizing effect wins, and the quarks reach their saturation point at a lower density than they would without the field. In other words, the magnetic field helps the transition to a new state of matter happen sooner. However, as the magnetic field gets even stronger, the squeezing effect takes over. The particles become so flattened that the quarks are spread out so thinly that they take longer to fill up the available space. Consequently, the density required to reach saturation actually increases again. This means that for very strong magnetic fields, the purely baryonic description of matter—where we treat the star as a gas of protons and neutrons—remains valid for a wider range of densities than it would in weaker fields.

The study confirms that the d-quark with a specific spin orientation is the first to reach this limit of fullness. This happens because the magnetic field pushes the particles into a state where this specific type of quark is most abundant. The research does not claim to have discovered a new state of matter or to have solved the mystery of neutron star interiors completely. Instead, it provides a precise calculation of when the old way of describing matter stops working. The findings suggest that the answer to "when does the transition happen?" depends entirely on how strong the magnetic field is. In weak fields, the transition comes early; in extreme fields, it is delayed. This insight helps refine our understanding of the dense matter inside neutron stars, showing that the magnetic environment plays a critical role in determining the limits of how much matter can be packed into a single point before the rules of physics change.

Technical Summary: Quark Saturation in Magnetized Pure Neutron Matter

Problem Statement
The paper addresses the transition from a hadronic description of dense matter to a quark-gluon description, specifically focusing on the phenomenon of "quark saturation." In dense baryonic matter, as density increases, low-momentum quark states eventually reach unit occupation probability. This limit, termed quark saturation, marks the breakdown of a purely baryonic description and the onset of a quark Fermi sea. While previous studies have estimated this onset to occur around 2–3 times the nuclear saturation density ( $n_0$ ), the precise threshold is sensitive to the internal structure and wave functions of baryons. This work investigates how strong, uniform magnetic fields ( $|eB| \sim \Lambda_{QCD}^2$ ) modify the internal structure of neutral $ddu$ baryons (neutrons and $\Delta^0$ -like states) and consequently alter the density at which quark saturation occurs.

Methodology
The author employs a simple non-relativistic constituent-quark model to describe individual baryons and constructs an ideal gas of these baryons in a static, uniform magnetic field. The methodology proceeds through several distinct steps:

Single-Baryon Spectra and Wave Functions:
- The model considers electrically neutral $ddu$ baryons. The Hamiltonian includes a harmonic confining potential and short-range color-electric and color-magnetic interactions (inspired by one-gluon exchange).
- The author introduces the pseudomomentum $\mathbf{K}$ to separate collective motion from internal dynamics. Using Jacobi coordinates, they derive the orbital energy spectra and wave functions.
- The spin-flavor structure is treated by diagonalizing the spin-dependent Hamiltonian (Zeeman and color-magnetic terms) in a basis connecting the zero-field neutron ( $n$ ) and $\Delta^0$ states.
- Numerical calculations are performed for three parameter sets (A, B, C) corresponding to different zero-field baryon radii ($0.50, 0.60, 0.70$ fm), with the constituent quark mass fixed at $m \simeq 0.33$ GeV.
Derivation of Single-Quark Distributions:
- From the fully symmetrized baryon wave function, the author constructs the one-body reduced density matrix for a single quark.
- This density matrix is diagonalized to find the natural basis states. The longitudinal distribution is obtained in momentum space, while the transverse distribution is expressed in the Landau-level basis.
- The analysis accounts for the magnetic translation symmetry and the mixing of Landau levels induced by the baryon's transverse pseudomomentum ( $K_\perp$ ).
Quark Occupation Probabilities in Dense Matter:
- An ideal gas of neutral baryons is assumed, where baryon states are filled up to a chemical potential $\mu_B$ .
- The quark occupation probability $f_i(n, p_z)$ for a specific flavor $i$ , Landau level $n$ , and longitudinal momentum $p_z$ is calculated by convoluting the baryon distribution function with the single-quark distributions derived from the baryon wave functions.
- Quark saturation is defined as the baryon density $n_B$ at which the maximum single-quark occupation probability reaches unity ( $f_{max} = 1$ ).

Key Results

Baryon Structure in Strong Fields:
- Spin-Flavor Polarization: As the magnetic field increases, the lowest energy branch ( $e_{n\downarrow}$ ) becomes increasingly polarized toward the configuration $d_\downarrow d_\downarrow u_\uparrow$ . In the strong-field limit, the neutron-like state evolves into a pure $d_\downarrow d_\downarrow u_\uparrow$ state.
- Transverse Squeezing: The magnetic field compresses the baryon in the transverse direction, significantly reducing the transverse size of the wave function.
- Dimensional Reduction: The transverse dispersion relation flattens, leading to a rapid increase in the transverse effective mass ( $M_\perp$ ). This indicates a suppression of transverse kinetic energy and an effective dimensional reduction of baryon dynamics.
Quark Saturation Density ( $n_{B}^{qsat}$ ):
- At finite magnetic fields, the $d_\downarrow$ quark in the lowest Landau level ( $n=0$ ) is the first to reach saturation.
- The saturation density exhibits a non-monotonic dependence on the magnetic field strength:
  - Weak Fields: $n_{B}^{qsat}$ decreases. This is driven by spin-flavor polarization, which enhances the population of the $d_\downarrow$ state, and the concentration of quarks into the lowest Landau level.
  - Strong Fields: $n_{B}^{qsat}$ increases. This is driven by transverse spatial squeezing. As the field strengthens, the internal wave function compresses, and the number of available guiding-center states per unit transverse area increases (scaling with $|eB|$ ). This broadens the quark transverse momentum distribution and reduces the occupation probability of any single transverse state, thereby delaying saturation.

Significance and Claims
The paper claims to provide a conceptual framework for understanding the interplay between hadronic structure and quark saturation in strong magnetic fields. The author emphasizes that while the field strengths considered ( $|eB| \sim \Lambda_{QCD}^2$ ) are substantially larger than typical magnetar surface fields and may exceed global equilibrium estimates, the study offers qualitative insights into how dense matter evolves toward the strong-field regime.

The primary contribution is the identification of a competition between two magnetic effects:

Spin-flavor polarization, which promotes saturation by concentrating quark weight into specific low-energy states.
Transverse spatial squeezing, which delays saturation by increasing the phase space density of available states (guiding centers) and broadening momentum distributions.

The author explicitly states that their results do not describe the state of matter beyond quark saturation, nor do they claim direct phenomenological application to current neutron star observations due to the extreme field strengths required. Instead, the work serves as a theoretical probe into the limits of baryonic descriptions and the mechanisms governing the transition to quark degrees of freedom. The author suggests that future work should extend this framework to include charged baryons, realistic interactions, finite temperature, and other baryonic species to apply these findings to realistic neutron star matter.

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