Unified Functional-Holographic Theory of the QCD Critical End Point

Abstract

We present a first-principles, multi-scale theory of the QCD critical end point. It unifies Dyson-Schwinger dynamics of quark propagation, functional renormalization-group evolution of the scale-dependent effective action, and Polyakov-Nambu-Jona-Lasinio thermodynamics for chiral and deconfinement order parameters. It also includes a holographic Maxwell-Chern-Simons sector that supplies baryonic and topological response within a single, renormalization group consistent framework. Within this construction the axial-anomaly channel is evolved rather than tuned. The holographic topological susceptibility feeds directly into the flow of the determinantal interaction. As a result, anomaly-induced flavor mixing weakens as deconfinement sets in. This behavior drives the chiral condensate and the Polyakov loop toward a self-dual fixed point. At this point their renormalizations coincide, their residual mixing vanishes, and a unified order parameter controls criticality. The theory is anchored to continuum-extrapolated lattice thermodynamics and conserved-charge fluctuations at vanishing baryon density. This anchoring is implemented through a lattice-calibrated Polyakov sector. The framework respects exact thermodynamic identities by enforcing stationarity at each scale. It also recovers the ideal-gas limit at high temperature and the Goldstone limit at low temperature. Solving the coupled Dyson-Schwinger, renormalization-group, and holographic equations yields the critical end point as an output rather than an assumption. The critical end point appears at temperatures around one hundred forty megaelectronvolts and baryon chemical potentials of a few hundred megaelectronvolts. The theory predicts a small positive curvature of the crossover line near zero density.

Figures (10)

• Figure 1: QCD phase diagram calculated from the unified DSE-FRG-V-QCD system: crossover line from the maximum of $-\partial_T\phi_l$ and schematic first-order continuation terminating at the CEP determined by $\partial_M^2\Omega=\partial_M^3\Omega=0$.
• Figure 2: Normalized chiral condensate and Polyakov loop at $\mu_B=0$ with parameters $(a_i,b_3,T_0)$ calibrated to lattice thermodynamics Fukushima:2008wgBorsanyi:2020.
• Figure 3: Dimensionless baryon susceptibility $\chi_B/T^2=\chi_2^B$ showing sharpening and shift with increasing $\mu_B$, computed from $\chi_B=\partial n_B/\partial\mu_B$ with full implicit $(\sigma,\Phi)$ feedback.
• Figure 4: Prediction for $\kappa\sigma^2=\chi_4^B/\chi_2^B$ versus $\sqrt{s_{NN}}$ along a continuous freeze-out trajectory $T_f(\mu_B)=T_c^{(0)}[1-\kappa_f(\mu_B/T_c^{(0)})^2-\lambda_f(\mu_B/T_c^{(0)})^4]$ with $\mu_B(\sqrt{s_{NN}})=A/(1+B\sqrt{s_{NN}})$. The nonmonotonic structure originates from proximity to the CEP in the $(T,\mu_B)$ plane.
• Figure 5: Pressure and energy density as functions of temperature at $\mu_B=0$, normalized by $T^4$, shown as smooth, color-coded curves.