QCD phase-transition under the light of Thermofractal

TL;DR

The paper addresses the mismatch between the GWW-model's sharp deconfinement transition and the lattice QCD observation of a smooth crossover in SU(3) with dynamical quarks. It introduces a thermofractal framework, employing a $q$-deformed derivative and $q$-exponential statistics, and derives $q$ from the QCD $eta$-function via $1/(q-1)=(11N_c-2N_f)/3$, linking fractal momentum-space to the running coupling. A unified master equation for the Polyakov loop $\langle L\rangle$ emerges, with a temperature-dependent variance $\sigma^2(T)\sim T^{1/(q-1)}$, predicting algebraic growth in the gluonic confining regime ($\langle L\rangle\propto T^{11}$) and rapid suppression in the deconfined regime ($1-\langle L\rangle\propto T^{-21}$), and reproducing lattice data with $\chi^2\approx1.12$. The framework provides a microscopic mechanism for partial deconfinement that reconciles matrix-model topology with non-extensive thermodynamics, identifies a distinct gluonic stage below $T_c$, and suggests extensions to PNJL models and heavy-ion observables for experimental tests.

Abstract

The deconfining transition in $SU(3)$ gauge theory, traditionally interpreted through the Gross-Witten-Wadia (GWW) model as a sharp third-order phase transition in the large-$N_c$ limit, appears as a smooth crossover in lattice QCD. This work demonstrates that the transition is topologically smoothed into a crossover by incorporating the fractal momentum space structure inherent to thermofractals. By matching the non-extensive $β$-function to one-loop QCD results, a fundamental scaling of the thermofractal index $q$ is derived as a function of the number of flavours $N_f$. It is proven that applying a $q$-deformed derivative operator $\mathcal{D}_q$ to the $q$-logarithm of the eigenvalue distance results in a non-extensive measure that effectively smears the topological stiffness of the gauge vacuum. A unified master equation for the Polyakov loop $\langle L \rangle$ is presented, governed by the thermofractal index $q$ and a single variance parameter $σ^2(T)$ that scales as $T^{1/(q-1)}$. The observed phase dynamics are shown to be asymptotic limits of this unified density: a ``soft'' algebraic growth $\langle L \rangle \propto T^{11}$ in the 1D string-like confined regime for $N_f=0$, and a rapid $1 - \langle L \rangle \propto T^{-21}$ suppression in the 3D deconfined volume for $N_f=3$. This approach provides a microscopic foundation for partial deconfinement theory and reproduces lattice QCD data with a reduced $χ^2 \approx 1.12$, offering a rigorous reconciliation between matrix model topology and the continuous QCD crossover.

Figures (1)

• Figure 1: Temperature dependence of the Polyakov loop $\langle L \rangle$ in $SU(3)$ gauge theory. The black circles represent $(2+1)$-flavor lattice QCD data Borsnyi2010. The solid blue line denotes the Two-Phase Thermofractal model, which incorporates a gluonic phase ($q=12/11$) below $T_c$ and a quark phase ($q=8/7$) above $T_c$. The solid green line represents the rigid Gross-Witten-Wadia (GWW) transition ($q=1$), which exhibits a sharp third-order discontinuity at $T_c$. The dashed red line indicates the pure quark phase ($q=8/7$) extrapolated to the vacuum, highlighting the necessity of the gluonic phase at low temperatures. The best fit parameters are: $\Lambda=0.141 \pm 0.002$ and $L_c=0.201 \pm 0.005$. The reduced $\chi^2$ is 1.12. Cross-validation with HotQCD data Bazavov2024 (red symbols, not used in calibration) yields similar agreement, confirming model independence.