Evidence for a Continuous Hadron--Quark Transition in Cold Dense Matter

Abstract

Statistical evidence for a continuous hadron-quark transition is found in this work. Confronting microscopic descriptions connecting the Parity Doublet Model and the Nambu--Jona-Lasinio model with observationally constrained non-parametric equations of state, Bayesian model comparison decisively favors a boundary-free crossover over the conventional first-order Maxwell construction ($Δ\ln\mathcal{Z} \sim +7.20$) and fixed-boundary crossover ($Δ\ln\mathcal{Z} \sim +5.16 $). The preferred crossover decouples the intermediate stiffening from the intrinsic stiffness of each phase, allowing both sectors to exhibit physical self-consistency. The chiral-invariant nucleon mass is large, $M_0 = 834^{+28}_{-92}\text{ MeV}$, as expected for a substantial baryon mass surviving chiral restoration. While the quark sector accommodates a small pairing gap $Δ_{\rm CFL} = 88^{+83}_{-59}\text{ MeV}$ consistent with perturbative QCD limits, avoiding the excessively large gaps $Δ_{\rm CFL} \gtrsim 200\text{ MeV}$ required by the other constructions. These results demonstrate that a realistic unified description represents a continuous transition deviating substantially from isolated effective models of hadrons and quarks. Exploring genuine phase boundaries is therefore necessary through the emergence of spinodal instabilities within unified frameworks.