Thermofractals Demonstrate Smooth QCD Phase-Transition, Scaling with Number of Flavours

The nature of the quark-gluon plasma transition, a fundamental process in the early universe and high-energy collisions, remains a key question in particle physics. Airton Deppman of the University of Sao Paulo, along with colleagues, present compelling evidence that this transition is not the sharp phase change predicted by some theoretical models, but rather a smooth crossover. Their research introduces the concept of thermofractals to explain how the complex, fractal structure of momentum space softens the transition between confined and deconfined states of quarks and gluons. By successfully matching their thermofractal approach to established one-loop quantum chromodynamics calculations, the team derives a crucial scaling relationship and offers a microscopic foundation for understanding partial deconfinement, ultimately reconciling theoretical predictions with lattice QCD data. This work provides a new and rigorous framework for investigating the dynamics of the quark-gluon plasma and its behaviour at extreme temperatures.

Thermofractals, QCD, and Vacuum Topological Stiffness

The research investigates the space structure inherent to thermofractals, exploring connections with quantum chromodynamics (QCD). By matching the non-extensive β-function to one-loop QCD results, a fundamental scaling relationship for the thermofractal index, denoted as ‘q’, is derived as a function of the number of flavours, ‘Nf’. The application of a q-deformed derivative operator, ‘Dq’, to the q-logarithm of the eigenvalue distance is demonstrated to yield a non-extensive measure. This measure effectively smears the topological stiffness of the gauge vacuum, offering a novel approach to understanding vacuum structure.

A unified master equation governing the Polyakov loop, ⟨L⟩, is presented, incorporating the thermofractal index ‘q’ and a single variance parameter, ‘σ2(T)’, which scales as ‘T 1/(q−1)’. This equation provides a consolidated framework for analysing the behaviour of the Polyakov loop in extreme conditions. The observed phase dynamics are then shown to be asymptotic limits of this unified density, described as a “soft” algebraic behaviour. This suggests a connection between thermofractal geometry and the phase transitions observed in QCD.

Thermofractals Smooth Deconfining Transition in SU(3) Theory

Scientists have demonstrated a topological smoothing of the deconfining transition in gauge theory, traditionally viewed as a sharp phase transition, by incorporating the inherent fractal structure of thermofractals. The research, focused on SU(3) gauge theory, reveals that this transition manifests as a smooth crossover, aligning with observations from lattice QCD calculations. By matching a non-extensive β-function to one-loop QCD results, the team derived a fundamental scaling for the thermofractal index, q, as a function of the number of flavours, Nf. Specifically, the scaling is defined as q − 1 = 11Nc − 2Nf / 3, establishing a direct link between the fractal geometry and the fundamental parameters of quantum chromodynamics.

Experiments revealed that applying a q-deformed derivative operator to the q-logarithm of eigenvalue distance generates a non-extensive measure that effectively reduces the topological stiffness of the gauge vacuum. This innovative approach led to the formulation of a unified master equation governing the Polyakov loop, a key order parameter for the quark-gluon plasma. The equation is governed by the thermofractal index q and a variance parameter, σ2(T), which scales as T 1/(q−1), demonstrating a temperature dependence directly linked to the fractal dimension. This scaling behaviour provides a microscopic foundation for understanding partial deconfinement, a phenomenon where the gauge group spontaneously breaks into confined and deconfined sectors.

Data shows that the observed phase dynamics represent asymptotic limits of the unified density, exhibiting a “soft” algebraic growth of the Polyakov loop proportional to T 11 in the 1D string-like confined regime when Nf = 0. Conversely, a rapid suppression of 1 −⟨L⟩ proportional to T −21 was recorded in the 3D deconfined volume for Nf = 3. The study successfully reproduces lattice QCD data with a reduced χ2 value of approximately 1.12, offering a rigorous reconciliation between matrix model topology and the continuous QCD crossover observed in experiments. The breakthrough delivers a novel framework for analysing the confined/deconfined phase transition, utilising the hierarchical structure of thermofractals and the associated Tsallis statistics. Measurements confirm that the thermofractal index q can be calculated from fundamental and adjoint representation parameters, indexing a fractal geometry within the momentum space. This work establishes that the smoothing of the transition is not merely a mathematical convenience, but a natural consequence of the fractal momentum space structure inherent in SU(N) gauge theories, offering a deeper understanding of the transition from hadronic matter to the quark-gluon plasma.

Fractal Momentum Space and QCD Deconfinement

This work establishes that the smooth crossover observed in SU(3) deconfinement is a direct consequence of a fractal structure within momentum space. By relating the thermofractal index to the beta-function of Quantum Chromodynamics, researchers have derived analytical scaling laws that accurately align with lattice QCD data, achieving a reduced chi-squared value of approximately 1.12. This approach provides a mathematical foundation for the partial deconfinement theory previously proposed by Hanada and colleagues, offering a cohesive framework for understanding the QCD vacuum. The observed “topological smoothing” is understood as the continuous development of deconfined SU(M) clusters, described by q-exponential statistics rather than the more conventional Gross-Witten-Wadia measures.

This reconciliation of matrix model topology with non-extensive thermodynamics delivers a unified description of the QCD vacuum across the deconfining transition. The authors acknowledge a limitation in that their current model focuses on the deconfining transition specifically, and future research could extend the application of thermofractals to chiral transitions, utilising PNJL models and exploring connections with experimental observations from heavy-ion collisions. Specifically, the research suggests a distinct gluonic stage exists below the critical temperature, potentially detectable through characteristic shifts in thermal photon emissivity and a specific temperature dependence in low-mass dilepton production rates. These signatures offer a potential avenue for probing the sub-critical self-organisation of the QCD vacuum, providing a testable link between theoretical modelling and experimental physics. Further investigation may also refine the understanding of non-perturbative corrections, with the current findings indicating a faster decay than typically observed in standard polynomial models.