Illustrating the liquid gas transition of nuclear matter in QCD

We demonstrate that the liquid-gas transition of nuclear matter can be rigorously described with the quantum chromodynamics by combining the quark gap equation and the Faddeev equation of nucleon. Our investigation focuses on this transition at zero temperature and finite chemical potential, revealing a finite difference between the gas and liquid solution of the quark propagator. This difference emerges from the shift of the nucleon pole mass in medium, which is generated in the nucleon channel of the quark gap equation. We prove that such a difference is precisely the contour contribution from the shift of the nucleon pole. The resulting discontinuity manifests as a first-order phase transition and fundamentally determines both the nuclear binding energy and the saturation density. We then derive an analytical relation between the binding energy and the sigma term of the nucleon, yielding a binding energy of $E/A=15.9\,\textrm{MeV}$. Furthermore, by establishing the relation between the nuclear saturation density and the vector charge of nucleon in association with the binding energy, we determine the saturation density to be $n_{\textrm{B}}^{0}=0.15\,\textrm{fm}^{-3}$.