Critical Bag Radius Overview

• Critical bag radius is a threshold value that defines qualitative transitions in confinement and stability across various physical and mathematical models.
• It quantifies the point where internal pressure, energy density, and boundary conditions combine to trigger phase changes, deconfinement, and chiral restoration.
• This concept is pivotal in diverse fields—from hadronic bag models and neutron star configurations to porous media and graph theory—with practical insights on system scaling and compactness.

A critical bag radius denotes a threshold value of spatial confinement in various physical, mathematical, and modeling contexts where the interplay between internal pressure, energy density, geometry, and boundary conditions leads to qualitative changes in system behavior. Its precise definition, role, and implications depend on the domain: in hadronic and stellar physics it directly governs confinement and stability; in statistical and continuum models it signals phase transitions or crossovers; and in mathematical models it encodes geometric or topological transitions. The following sections detail the key concepts, implementations, and theoretical significance associated with the critical bag radius across representative areas.

1. Definition and Thermodynamic Role in Hadronic Bag Models

In the MIT bag model and its generalizations, the critical bag radius is fundamentally linked to color confinement. Here, quarks are treated as free, relativistic particles inside a finite region ("bag") of radius $R$, with boundary conditions enforcing confinement. The energy of the system comprises the quark kinetic energy and the external "bag pressure" $B$ due to vacuum effects. A critical bag radius emerges as a structural boundary: it is the value of $R$ below which the bag binding energy $E_B$ turns negative,

$E_B = \frac{4}{3}\pi R^3 B - Z_0 R,$

where $Z_0$ is a zero-point energy parameter (Liu et al., 22 Jan 2025). For $R < R_c$, $E_B<0$ signals a configuration that is genuinely bound and exhibits "compact intention," qualifying as a compact hadron or multiquark state. The canonical value derived by variational minimization is $$R_c = 5.615\, \mathrm{GeV}^{-1} \approx 1.11\, \mathrm{fm}$$ for typical bag model parameters.

The bag constant $B$ itself is defined via

$B = \frac{M-E_q}{V},$

where $M$ is the total mass (energy) of the hadron and $E_q$ is the quark energy, further partitioning the total energy into quark and gluon contributions (Tan et al., 2010). The radius may also be linked to the internal temperature $T$ via a scaling relation, e.g.,

$R T^{4/3} = 0.0524$

demonstrating that as the system is compressed ($R$ decreases), $T$ rises, ultimately leading toward deconfinement at critical conditions (Tan et al., 2010).

2. Bag Radius in Phase Transitions, Deconfinement, and Chiral Restoration

The critical bag radius delineates the phase boundary for quark deconfinement and chiral symmetry restoration in finite-temperature QCD-inspired models. Within the global color symmetry model (GCM), as the temperature approaches the critical temperature $T_c$, the effective bag constant and nucleon mass decrease, while the bag radius $R(T)$ diverges,

$$\mathcal{B}(T_c) \to 0, \quad M(T_c) \to 0, \quad R(T_c) \to \infty$$

(Mo et al., 2010). This transition corresponds to the dissolution of the bag (confinement loss) and the disappearance of the chiral condensate, signaling both deconfinement and chiral symmetry restoration.

In color confinement and QCD phase diagrams, the surface tension of the bag also plays a crucial role, entering the string tension expression for cylindrical bags (Bugaev et al., 2011):

$$\sigma_{\text{str}}(T) = 2\pi R\, \sigma_{\text{surf}}(T) - \pi R^2 p_v(T) + \frac{T\tau}{L} \ln\frac{\pi R^2 L}{V_0}$$

Approaching the endpoint or crossover, $\sigma_{\text{str}}\to 0$, driving $R\to\infty$, indicating bag destabilization. Negative surface tension in the crossover regime facilitates the continuous phase transition.

3. Critical Bag Radius in Compact Stars and Neutron Star Models

In astrophysical applications, particularly the structure of neutron stars and hybrid stars, the critical bag radius becomes a marker for the stability and maximum compactness of self-bound configurations. In these contexts, the critical radius is controlled by the bag constant $B$ (often with density or chemical potential dependence):

• A larger $B$ results in smaller, more compact stars.
• The mass-radius ($M$-$R$) relation often obeys $M\propto B^{-1/2}$ and $R \propto B^{-1/2}$ (Li et al., 2010), determining the critical configuration limits.

In advanced models, a density-dependent bag constant

$$\mathcal{B}(n) = \mathcal{B}_\infty + (\mathcal{B}_0 - \mathcal{B}_\infty) \exp\left[-\beta (n/n_0)^2\right]$$

(Yazdizadeh et al., 2013, Sen et al., 2021) more realistically represents the changing vacuum pressure with baryon density, affecting the phase transition threshold and raising maximum masses while reducing radii, i.e. compactifying the star.

In strange star models using the Finch–Skea metric, the critical (maximum allowed) bag radius $b_{\max}$ for a given $B$ is constrained by stability, causality, and real metric conditions. Exceeding $b_{\max}$ results in loss of physical solutions (Das et al., 2023).

4. Scaling Relations, Spectral, and Transition Theory

The critical bag radius is generalized in several other contexts as a scaling variable governing phase transitions or geometric/topological crossovers. For a mean-field polymer model at near-critical temperature, the radius transitions according to

$$r(B, t) \sim \begin{cases} (B - B_{\text{cr}})^{-1} & \text{if}\ (B - B_{\text{cr}})\sqrt{t} \geq 1 \ \sqrt{t} & \text{if}\ (B - B_{\text{cr}})\sqrt{t} \leq 1 \end{cases}$$

(Koralov et al., 2020), where $B_{\text{cr}}$ represents the critical threshold for a collapse (bag-like state) vs. diffusive expansion.

In super-Brownian motion, the largest empty ball radius at large times asymptotically behaves as

$$\lim_{t\to\infty}\mathbb{P}\left(\frac{R_t}{t^{(1/d)\wedge(3-d)^+}}\geq r\right)=e^{-A_d(r)}$$

(Xiong et al., 2022), with $A_d(r)$ a large deviation function capturing emptiness in the critical process.

5. Geometry, Percolation, and Pore Space Statistical Models

A formal analog appears in porous media: the critical pore radius $\delta_c$ is defined by the void percolation threshold, marking the largest probe radius that can traverse the sample (Klatt et al., 2021). It is tightly correlated with the second moment of the pore-size distribution,

$\delta_c^2 \propto \langle \delta^2 \rangle,$

which, in turn, links to fluid permeability estimates via

$$k = \mathcal{L}^2 / \mathcal{F}, \quad \mathcal{L}^2 \approx \langle \delta^2 \rangle$$

This identifies the critical pore radius as the geometric percolation boundary governing transport.

In colloidal systems, the critical radius $R_c$ determines the character of bridging transitions: for spherical particles, transitions are first-order only for $R > R_c$ and rounded for $R < R_c$ (Malijevský et al., 2015).

6. Structural Graph Theory and Factor-Critical Concepts

In spectral graph theory, a metaphorical "critical bag radius" relates to robustness thresholds in factor-critical graphs. A graph is $k$-factor-critical if removal of any $k$ vertices leaves a perfect matching. Theorem 3.1 gives a sharp spectral radius bound:

$p(G) > \rho(n,k)$

makes $G$ $k$-factor-critical, where $\rho(n, k)$ is the root of a cubic polynomial in $n$ and $k$ (Zhou et al., 2023). While not a direct geometric radius, this spectral bound serves as a "global radius" governing the robustness of local vertex "bags" against perfect matching loss.

7. Implementation in Dynamical and Chiral Bag Models

Modern extensions incorporate chiral dynamics, e.g., using dynamical bag functions tied to the nonlinear pion field, with the "critical bag radius" defined as the point $R$ where the chiral profile achieves $F(R)=\pi/2$ (Jia et al., 2013). This marks the transition zone between the quark core and the pion cloud, with static properties such as proton charge radius and magnetic moment directly set by the dynamical (non-singular) bag boundary.

Summary Table: Critical Bag Radius Across Contexts

Domain	Definition/Role	Threshold Behavior/Effect
MIT Bag Model	Radius where $E_B<0$ for binding	Compactness of hadronic or multiquark states
Stellar Structure	$M,R \propto B^{-1/2}$; $b_{\max}$ for physical solutions	Maximum allowed compact star size
QCD/Crossover	$R\to\infty$ as deconfinement ($\mathcal{B}(T_c)\to 0$)	Onset of deconfinement/chiral symmetry
Polymer Physics	$(B-B_{\text{cr}})^{-1}$ collapse scaling	Swollen-to-collapsed transition
Porous Media	Max probe radius at percolation threshold ($\delta_c$)	Onset of sample-spanning flow
Graph Theory	Minimal spectral radius for factor-criticality ($\rho(n,k)$)	Graph robustness against matching loss

Concluding Remarks

The critical bag radius functions as a parameter signifying phase boundaries, geometric percolation, stability, and compactness in systems governed by interplay between energy, pressure, connectivity, and boundary conditions. Its explicit value and functional role depend on the physical, statistical, or mathematical formulation, but the cross-cutting concept remains that at or below this critical threshold, qualitative changes manifest—ranging from stable compact hadrons or stars, to topological transitions in polymers or porous media, to connectivity resilience in graphs.