Mechanical properties of the nucleon in the chiral confining model. II -- in-medium evolution of the nucleon properties

This paper investigates the evolution of nucleon properties within nuclear matter using the chiral confining model, revealing how the interplay of confinement and chiral symmetry breaking drives mass modifications and repulsive three-body forces essential for nuclear saturation and neutron star equations of state.

The Gist

Imagine the atomic nucleus not as a collection of hard, solid marbles, but as a bustling city where the "citizens" (protons and neutrons, or nucleons) are actually complex, squishy balloons filled with smaller, energetic particles called quarks. These balloons are wrapped in a fuzzy, vibrating cloud made of even smaller particles called pions.

This paper is the second part of a study by physicists Guy Chanfray, Hubert Hansen, and Bikram Keshari Pradhan. Their goal is to understand what happens to these "nucleon balloons" when they are squeezed together inside a dense crowd (like inside an atomic nucleus or the core of a neutron star).

Here is the breakdown of their work using simple analogies:

1. The Setup: The "Squeezed Balloon" Model

The authors use a model called the Chiral Confining Model.

• The Balloon (The Nucleon): Inside the nucleus, a nucleon is like a balloon held together by a string-like force (confinement) that keeps the quarks from flying apart.
• The Fuzzy Cloud (The Pion Cloud): Surrounding the balloon is a fuzzy cloud of pions. This cloud is crucial because it acts like a cushion or a shock absorber.
• The Squeeze (The Scalar Field): When you put these balloons into a crowded room (nuclear matter), they feel a "pressure" from the crowd. In physics terms, this is a "scalar field." It's like the air pressure in a room increasing, which tries to shrink the balloons.

2. The Problem: Why Don't Nuclei Collapse?

In the past, scientists had a puzzle. If you squeeze these balloons too hard, the "cushion" (the pion cloud) should get squished, making the attraction between balloons stronger. This should cause the whole nucleus to collapse in on itself. But in reality, nuclei are stable; they don't collapse.

The authors propose a solution: The balloons fight back.
When the crowd squeezes the balloon, the balloon doesn't just shrink passively. The internal structure changes. The quarks inside rearrange themselves, and the fuzzy pion cloud starts to "evaporate" or thin out. This reaction creates a repulsive force (a push back) that balances the squeeze. This push-back is what keeps the nucleus stable and prevents it from collapsing.

3. The Method: The "Stability Test"

To figure out exactly how the balloon behaves, the authors used a rule called the von Laue stability condition.

• The Analogy: Imagine a balloon floating in the air. For it to be stable, the air pushing out from the inside must perfectly balance the air pushing in from the outside. If the inside pressure is too high, it pops; if it's too low, it shrivels.
• The Application: The authors calculated the internal "pressure" of the nucleon (from the quarks) and the "pressure" from the pion cloud and the confining strings. They adjusted the size of the nucleon until these forces balanced perfectly. This allowed them to find the "true" size and mass of a nucleon inside a nucleus.

4. The Discovery: What Happens Under Pressure?

The paper presents two main scenarios:

Scenario A: The "Static" Nucleon (The Localized Bag)
They first looked at a nucleon that is stuck in one spot.

• Result: As the "squeeze" (scalar field) gets stronger, the nucleon gets slightly larger, and the fuzzy pion cloud gets thinner. The energy inside spreads out. It's like a sponge soaking up water but then slowly drying out and expanding as the pressure changes.

Scenario B: The "Moving" Nucleon (The Physical Nucleon)
They then looked at a nucleon that is moving freely (which is more realistic).

• Result: They found that the nucleon's mass actually stays relatively stable or even gets slightly heavier as the squeeze increases, up to a certain point.
• The "Evaporation": The most striking finding is that as the density increases, the fuzzy pion cloud "evaporates." The nucleon starts to look less like a fuzzy balloon and more like a bare bag of quarks.
• The Sweet Spot: The nucleon is most stable at a specific level of squeezing. If you squeeze it too hard (beyond a certain density), the nucleon can no longer maintain its structure as a distinct object.

5. Why This Matters for Neutron Stars

The authors connect this to neutron stars, which are the densest objects in the universe.

• The Analogy: Imagine a neutron star as a giant pile of these squeezed balloons.
• The Prediction: As you go deeper into the star, the pressure gets so high that the "fuzzy clouds" of the nucleons disappear. The star transitions from being made of "fuzzy balloons" to being made of "bare bags" of quarks packed tightly together.
• The "Hard" Matter: This transition creates a very stiff, hard material (called "hard deconfined matter"). This stiffness is important because it determines how heavy a neutron star can get before it collapses into a black hole.

Summary of the Main Takeaways

1. Nucleons are flexible: They aren't hard rocks; they are complex structures that change shape and size when squeezed.
2. The "Evaporation" Effect: Under high pressure, the fuzzy cloud surrounding the nucleon disappears, leaving a denser core.
3. Stability comes from balance: The stability of nuclear matter relies on a delicate balance between the internal pressure of the quarks and the pressure from the pion cloud.
4. New Map for Neutron Stars: By understanding how these "balloons" behave under pressure, the authors have created a new map for the equation of state (the rules of pressure and density) inside neutron stars, suggesting a phase where matter becomes a "hard" collection of quark cores.

In short, the paper uses the physics of a "squishy, fuzzy balloon" to explain why atomic nuclei don't collapse and what happens to matter when it is crushed to the limits of the universe.

Technical Summary

Technical Summary: Mechanical Properties of the Nucleon in the Chiral Confining Model II

Problem Statement
This work addresses the evolution of nucleon properties when bound within nuclear matter, specifically focusing on the interplay between the nuclear many-body problem and nucleon internal structure. A central challenge in chiral effective theories is the nuclear saturation mechanism. Standard approaches often suffer from instability due to attractive three-body forces generated by the $s^3$ tadpole diagram in the chiral effective potential (associated with the scalar field $s$). While the Quark-Meson Coupling (QMC) model and the Chiral Confining Model (CCM) propose that the nucleon's response to the scalar field generates repulsive three-body forces to counteract this attraction, a rigorous microscopic derivation of these response parameters ($g_S$ and $\kappa_{NS}$) and the mechanical stability of the in-medium nucleon has remained incomplete. Specifically, previous applications of the CCM lacked a fully satisfactory treatment of the nucleon's mechanical stability in the presence of both a confining potential and a surrounding pion cloud.

Methodology
The authors extend the formal developments of a companion paper (labeled "I") to the case of a bound nucleon within the CCM framework. The model combines a non-perturbative confining interaction (derived from the Field Correlator Method, FCM, and modeled as a string-like potential) with a chiral pion cloud coupled to constituent quarks.

Key methodological steps include:

1. Construction of Trial States: The study utilizes two types of trial states for the in-medium nucleon:
   • A localized static bag ($|N(X_N)\rangle$), where the center of mass is fixed, and the wave function is a product of three quark orbitals.
   • A momentum-projected state ($|N(P_N)\rangle$), which projects the localized state onto a well-defined total three-momentum to describe a physical nucleon.
2. Effective Action and Lagrangian: Starting from an effective action derived within the FCM framework, the authors construct a local effective Lagrangian. This includes the coupling of constituent quarks (with in-medium mass $S$) to a pion field with finite spatial extent. A non-local coupling is introduced to avoid the collapse of the quark core caused by divergent attractive pion-quark interactions.
3. Mechanical Stability (von Laue Condition): A critical component of the methodology is the imposition of the von Laue stability condition. This condition requires the integral of the local pressure distribution to vanish ($\int d^3r \langle \hat{P}(r) \rangle = 0$). This condition is used to self-consistently fix the size parameter ($b$) of the Gaussian quark wave function for a given nuclear scalar field strength.
4. Energy-Momentum Tensor (EMT): The authors define the static EMT tensor and decompose it into local energy density ($\varepsilon(r)$) and mechanical pressure ($p(r)$) distributions. These distributions account for contributions from quark kinetic energy, confinement, pion-quark interactions, and the pion cloud itself.

Key Contributions

• Self-Consistent Determination of Response Parameters: By enforcing the von Laue stability condition, the paper derives the nucleon's scalar coupling constant ($g_S$) and the dimensionless scalar susceptibility ($C$) directly from the microscopic structure of the nucleon in a scalar field, rather than treating them as phenomenological inputs.
• Refined Nucleon Modeling: The work distinguishes between a "localized static bag" and a "momentum-projected physical nucleon," demonstrating how the latter provides a more realistic description of the in-medium evolution, particularly regarding the nucleon mass and size.
• Internal Mechanical Structure: The paper provides a detailed formalism for calculating and visualizing the internal energy density and pressure distributions of the nucleon in medium, explicitly separating quark and pion contributions.
• Equation of State (EoS) Mapping: The study proposes a mapping between the internal mechanical properties of the nucleon (specifically the "cloudy quark core") and the Equation of State of dense matter, relevant for the deep interiors of neutron stars.

Results

• In-Medium Evolution: As the nuclear scalar field $|s|$ increases (corresponding to higher density and partial chiral symmetry restoration), the constituent quark mass decreases. Consequently, the nucleon size parameter $b$ increases, indicating a dilution of the quark core.
• Pion Cloud "Evaporation": The analysis reveals a progressive "evaporation" or dilution of the pion cloud as density increases. The negative pion pressure, which balances the positive Fermi pressure in vacuum, decreases significantly in magnitude. This leads to a stiffening of the nucleon's internal equation of state.
• Response Parameters:
  • For the localized static bag, the extracted parameters are $g_S \approx 4.42$ and $C \approx 0.33$.
  • For the momentum-projected physical nucleon (using an effective string tension $\sigma_E = 0.08$ GeV to achieve a realistic mass), the extracted parameters are $g_S = 6.21$ and $C = 0.42$. These values are consistent with those used in previous phenomenological studies of nuclear matter saturation and are compatible with lattice QCD constraints on chiral extrapolations.
• Mass Evolution: The in-medium nucleon mass $M_N^*(s)$ exhibits a minimum at $|s|/F_\pi \sim 0.55$. Below this point (normal nuclear densities), the mass decreases with density; beyond this point, the concept of a nucleonic solution becomes unstable.
• Neutron Star Implications: The study suggests that at high densities, the "cloudy quark core" behaves as a hard deconfined matter phase. The mapping of the internal pressure to the bulk EoS indicates a breakdown density around $\rho_N \approx 0.8$ fm$^{-3}$, where the local energy density reaches $\sim 1$ GeV/fm$^3$. This density is lower than that predicted by NJL-soliton models, suggesting a stiffer EoS for dense matter.

Significance
The paper claims that its primary significance lies in providing a self-consistent, microscopic foundation for the response parameters ($g_S$ and $C$) that govern nuclear saturation. By utilizing the von Laue mechanical stability condition, the authors resolve the delicate balance between the Fermi pressure of quarks and the negative pressure of the pion cloud and confinement. This approach validates the use of the Chiral Confining Model to link low-energy QCD properties (chiral symmetry breaking and confinement) directly to nuclear many-body phenomena. Furthermore, the detailed analysis of the internal pressure and energy density distributions offers a new perspective on the transition from nuclear matter to deconfined quark matter in the cores of neutron stars, suggesting a "hard" deconfined phase intermediate between ordinary nuclear matter and pure quark matter.