Anomalous Thermal Dimension and the Enthalpy Renormalization Group Flow in QCD

A precise characterization of the QCD phase transition remains a fundamental open problem, primarily due to the intrinsically non-perturbative nature of the dynamics that govern the breakdown of (approximate) scale invariance in the vicinity of the critical temperature $T_c$. In this work, we formulate a thermodynamic scaling framework for the QCD enthalpy by deriving a linear Callan-Symanzik-type partial differential equation that governs its scale dependence. By mapping macroscopic observables onto a dimensionless auxiliary field space, we define an anomalous thermal dimension, $h(T)$, which probes the trace anomaly and the deformation of the conformal thermodynamic state-space geometry. This framework is confirmed against first-principles Lattice QCD data from the Wuppertal-Budapest collaboration, successfully capturing the localized "scaling bump" associated with the deconfinement crossover, a feature typically missed by analytical models such as the MIT Bag Model. Comparison with known universality classes shows that the enthalpy-based renormalization group flow is highly sensitive to explicit scale breaking from finite quark masses. These results indicate that $h(T)$ acts as a thermodynamic susceptibility to scale transformations, offering a new link between quantum scale anomalies and the information-geometric curvature of strongly interacting matter.