Regular black hole solutions and the quark chemical potential at the QCD phase transition

Technical Summary: Regular Black Hole Solutions and the Quark Chemical Potential at the QCD Phase Transition

Problem Statement
The paper investigates whether Quantum Chromodynamics (QCD)-inspired matter at finite quark chemical potential ($\mu$) can dynamically support the formation of regular black hole (RBH) interiors during gravitational collapse. While observational data from LIGO/Virgo and the Event Horizon Telescope (EHT) have constrained deviations from the Kerr paradigm, the theoretical question remains whether realistic high-density equations of state (EoS) can naturally yield non-singular compact objects with de Sitter-like cores, or if such regularity requires ad hoc imposition. Specifically, the authors examine if finite $\mu$ alters the thermodynamics of collapse sufficiently to prevent curvature singularities at the center.

Methodology
The authors couple two distinct effective QCD-inspired equations of state to a spherically symmetric, advanced Eddington–Finkelstein (EF) geometry:

1. Chiral Quark Model: A three-flavor model at finite temperature ($T$) and chemical potential ($\mu$), incorporating a scalar-pseudoscalar chiral field and a bag constant $B$.
2. Cold-QGP Mean-Field Model: A relativistic mean-field description where the gluon field is split into soft and hard modes, with hard modes acquiring a dynamical mass $m_g$.

Rather than assuming a regular mass profile $M(v, r)$ a priori, the authors reconstruct the mass function directly from the local conservation of energy-momentum ($T^{\mu\nu}_{;\nu} = 0$) and the Einstein equations. The matter source is treated as an effective anisotropic fluid. The analysis proceeds by:

• Deriving the radial temperature profile $T(r)$ from the conservation equation for each EoS.
• Integrating the energy density to reconstruct the mass function $M(v, r)$.
• Analyzing the near-center ($r \to 0$) scaling behavior of $M(v, r)$ and curvature invariants (Ricci and Kretschmann scalars) to test for regularity. A regular core requires $M(v, r) \sim r^3$ (de Sitter-like), ensuring finite curvature.

Key Contributions and Results

• Chiral Model Analysis:
  The conservation equation for the chiral model admits an exact solution in terms of the Lambert $W$ function. However, for physical coefficients relevant to the high-density QGP regime, the principal branch ($W_0$) yields a temperature scaling $T^2 \propto r^{-\zeta/\xi}$ where $\zeta/\xi \approx 1.86$. Consequently, the energy density diverges as $\rho \sim r^{-2\zeta/\xi}$, leading to a mass function correction scaling as $M - M_0 \propto r^{3 - 2\zeta/\xi} = r^{-115/159}$. This power law is singular (diverging as $r \to 0$) and incompatible with the $M \sim r^3$ condition required for a regular center. The alternative branch ($W_{-1}$) drives $T \to 0$, which lies outside the assumed high-temperature collapse regime.

• Cold-QGP Model Analysis:
  In the cold-QGP model, the implicit relation between radius and temperature drives $T \to \infty$ as $r \to 0$. The resulting energy density scales as $\rho \sim T^4 \propto r^{-4\delta/\gamma}$. The reconstructed mass function scales as $M - M_0 \propto r^{3 - 4\delta/\gamma}$. For the entire physical parameter range ($2/3 \le \delta/\gamma < 1$), the exponent $n = 3 - 4\delta/\gamma$ is strictly less than 3. Depending on the specific parameters, the mass function either diverges as a negative power of $r$, diverges logarithmically (at the boundary $\delta/\gamma = 3/4$), or vanishes slower than $r^3$. In all cases, the curvature invariants diverge, failing the regularity criterion.

• No-Go Result:
  The central finding is a restricted "no-go" theorem: within the effective frameworks of the chiral and cold-QGP models considered, finite quark chemical potential reshapes the thermodynamics of the collapse phase but does not by itself provide a self-regularizing black hole core. The stiff equation of state near the center prevents the necessary vacuum-like condition ($p \simeq -\rho$) from emerging dynamically.

• Proposed Regular Completion:
  To achieve a regular black hole, the authors propose an effective completion where the singular QCD phase is matched to an inner vacuum-like core ($p \simeq -\rho$) below a transition radius $r_c$. Using a smooth interpolating function, they demonstrate that such a matched geometry yields a regular de Sitter core ($M \sim r^3$), finite curvature invariants, and a double-horizon structure (inner and outer horizons) characteristic of RBHs. The exterior geometry remains indistinguishable from the Schwarzschild solution for $r > r_c + \Delta$.

Significance and Claims
The paper claims to provide a rigorous test of whether QCD-inspired matter at finite $\mu$ can dynamically resolve the singularity problem in gravitational collapse. The authors conclude that while finite $\mu$ modifies the collapse dynamics, the specific effective EoSs studied are insufficient to generate a regular core without additional short-distance physics.

The significance of the work lies in:

1. Dynamical Reconstruction: Moving beyond ansatz-based regular metrics to reconstruct the mass function from conservation laws and specific microphysical EoSs.
2. Limitation of QCD EoSs: Establishing that standard effective QCD descriptions (chiral and cold-QGP) are too "stiff" near the center to satisfy the $M \sim r^3$ regularity condition.
3. Necessity of New Physics: Arguing that any regular completion must invoke an additional inner phase (e.g., vacuum-like, color superconductivity, or non-perturbative effects) beyond the two QCD-inspired EoSs considered.
4. Phenomenological Framework: Providing a concrete matched-metric model (Sec. VII) that can be used to study observational signatures (shadows, lensing, quasinormal modes) of regular black holes, acknowledging that the inner core structure is currently an effective assumption rather than a derived result from first-principles QCD.

The authors explicitly state that their analysis is a "restricted no-go result" and not a complete model of QCD-driven regular black hole formation, emphasizing that future work must derive the inner vacuum-like phase dynamically from ultraviolet completions or specific microphysical mechanisms.