Comparative analysis of critical regions: The renormalized quark-meson model under Polyakov loop, quark back-reaction, and vector interaction effects

Technical Summary: Comparative Analysis of Critical Regions in Renormalized Quark-Meson Models

Problem Statement
The paper addresses the mapping of critical regions surrounding the Critical End Point (CEP) in the QCD phase diagram ($\mu$-$T$ plane). While Lattice QCD establishes a crossover at zero baryon density, effective theories predict a first-order transition at high density and low temperature, necessitating a CEP where these phases meet. Previous studies using Quark-Meson (QM) and Polyakov-Quark-Meson (PQM) models have often relied on inconsistent parameter-fixing schemes, specifically using curvature masses (derived from the second derivative of the effective potential at zero momentum) to fix parameters in models that include quark one-loop vacuum fluctuations. This approach is theoretically inconsistent because curvature masses only account for vacuum corrections at zero momentum, whereas physical meson properties are defined by pole masses. Furthermore, the impact of quark back-reaction in the Polyakov loop potential and repulsive vector interactions on the size and shape of critical fluctuations remains to be systematically quantified within a consistently renormalized framework.

Methodology
The authors employ the 2+1 flavor Quark-Meson (QM) model and its Polyakov-loop enhanced variants (PQM), specifically utilizing:

1. Renormalized Models (RQM/RPQM): The study utilizes the on-shell renormalized Quark-Meson (RQM) and Polyakov-loop enhanced RQM (RPQM) models. Parameters are fixed by matching counter-terms in the on-shell (OS) scheme with the modified minimal subtraction (MS) scheme, directly relating mass parameters and running couplings to physical pole masses ($m_\pi, m_K, m_\eta, m_{\eta'}, m_\sigma$) and decay constants ($f_\pi, f_K$). This ensures a consistent treatment of quark one-loop vacuum fluctuations.
2. Polyakov Loop Potentials: Two forms of the Polyakov loop potential are compared:
   • The Logarithmic (Log) form, which lacks quark back-reaction.
   • The PolyLog-glue form (improved from Ref. [46]), which incorporates quark back-reaction effects on the gluon sector.
3. Vector Interactions: Repulsive vector interactions are introduced via Yukawa coupling to vector mesons ($\omega, \phi$), with the coupling strength $g_\omega$ varied from 0.0 to 2.75.
4. Critical Region Mapping: The critical regions are mapped by computing contours of the normalized quark number susceptibility ratio, $R_q = \chi_q / \chi_q^{free}$, for values of 2, 3, and 5.
5. Chiral Limits: Calculations are performed for both the physical point ($m_\pi = 138$ MeV) and the light chiral limit ($m_\pi = 0$), utilizing large-$N_c$ chiral perturbation theory inputs to determine parameters in the chiral limit.

Key Contributions and Results

• Impact of Consistent Renormalization:

  • In the RQM model, the consistent on-shell treatment leads to a significantly stronger 't Hooft coupling ($c$) and weaker explicit chiral symmetry breaking strengths ($h_x, h_y$) compared to models using curvature masses.
  • Consequently, the CEP shifts to higher temperatures and lower chemical potentials compared to curvature-mass-based models (QMVT/PQMVT). For $m_\sigma = 500$ MeV, the RQM CEP is located at $(\mu_{CEP}, T_{CEP}) \approx (265, 39)$ MeV, whereas the QMVT model places it at $(286, 32)$ MeV.
  • The critical regions in the RQM model are broad, smooth, and symmetric, contrasting with the "neck-like" narrow structures found in curvature-mass-based models below the CEP.

• Role of Polyakov Loop and Back-Reaction:

  • Log RPQM (No Back-Reaction): The inclusion of the Log Polyakov loop potential shifts the CEP to higher temperatures (e.g., $T_{CEP} \approx 94.6$ MeV for $m_\sigma=500$ MeV) but compresses the critical region in the temperature direction, creating a pinched contour shape similar to previous PQMVT findings.
  • PolyLog-glue RPQM (With Back-Reaction): Incorporating quark back-reaction via the PolyLog-glue potential smooths the critical region. The contours become broader and more symmetric, with a significant expansion in the temperature direction compared to the Log RPQM case. The back-reaction effectively links the confinement-deconfinement and chiral transitions even at lower temperatures.

• Tricritical Point (TCP) Proximity:

  • In the light chiral limit ($m_\pi=0$), the study locates the Tricritical Point (TCP). For $m_\sigma = 500$ MeV, the TCP lies close to or inside the boundary of the $R_q=2$ critical region, suggesting it influences critical fluctuations around the CEP. For $m_\sigma = 400$ MeV, the TCP is located further away, implying negligible influence on the CEP fluctuations.

• Vector Interaction Effects:

  • Increasing the vector coupling strength $g_\omega$ weakens the first-order transition.
  • CEP Survival: The CEP and first-order transition survive up to a robust coupling of $g_\omega = 2.79$ (corresponding to a vector-to-scalar ratio $r \approx 0.86$). Beyond this value, the entire phase diagram becomes a crossover.
  • Critical Region Morphology: As $g_\omega$ increases, the critical regions shrink, particularly along the chemical potential axis. In the Log RPQM model, the contours become vertically oriented (elongated in $T$, compressed in $\mu$). The PolyLog-glue model moderates this compression, maintaining broader contours.

Significance and Claims
The paper claims that the consistent on-shell renormalization of quark one-loop vacuum fluctuations is crucial for obtaining physically realistic critical regions. The study demonstrates that previous models using curvature masses may overestimate the size of critical regions and misplace the CEP due to inconsistent parameter fixing.

The authors emphasize that their framework, particularly the RPQM model with PolyLog-glue potential, yields a CEP location ($T \sim 70-95$ MeV, $\mu \sim 227-252$ MeV) that aligns more closely with recent theoretical consensus and lattice QCD extrapolations than curvature-mass-based models.

Furthermore, the finding that the CEP survives vector couplings up to $g_\omega = 2.79$ is highlighted as having significant implications for the Equation of State (EoS) of compact stars. This robustness suggests that a first-order phase transition and a CEP could exist in the dense matter cores of neutron stars even with strong repulsive vector interactions, a result that contrasts with some NJL/PNJL studies where the transition disappears at lower coupling strengths. The work provides a refined mapping of critical fluctuations, essential for interpreting experimental signals from heavy-ion collisions (such as the Beam Energy Scan) and for constraining astrophysical models of dense matter.