Energy Density Functional of Confined Quarks: an Improved Ansatz

Abstract: Density Functional Theory (DFT) is a robust framework for modeling interacting many-body systems, including the equation of state (EoS) of dense matter. Many models, however, rely on energy functionals based on assumptions that have not been rigorously validated. We critically analyze a commonly used ansatz for confinement, where the energy functional scales with density as $U \propto n^{\frac{2}{3}}$. Our findings, derived from a systematic non-local energy functional, reveal that this scaling does not capture the dynamics of confinement. Instead, the energy functional evolves from $n^2$ at low densities to $n$ at high densities, governed by an infrared cutoff. These results suggest that models relying on such assumptions should be revisited to ensure more reliable EoS construction.