Quark and Nucleon Self-Energy in Dense Matter
  In a recent work we introduced a nonlocal version of the Nambu--Jona-Lasinio(NJL) model that was designed to generate a quark self-energy in Euclidean space that was similar to that obtained in lattice simulations of QCD. In the present work we carry out related calculations in Minkowski space, so that we can study the effects of the significant vector and axial-vector interactions that appear in extended NJL models and which play an important role in the study of the $\rho$, $\omega$ and $a_1$ mesons. We study the modification of the quark self-energy in the presence of matter and find that our model reproduces the behavior of the quark condensate predicted by the model-independent relation $<\bar qq>_{\rho} = <\bar qq>_0(1-\sigma_N\rho_N/f_{\pi}^2m_{\pi}^2 +...)$, where $\sigma_N$ is the pion-nucleon sigma term and $\rho_N$ is the density of nuclear matter. (Since we do not include a model of confinement, our study is restricted to the analysis of quark matter. We provide some discussion of the modification of the above formula for quark matter.) The inclusion of a quark current mass leads to a second-order phase transition for the restoration of chiral symmetry. That restoration is about 80% at twice nuclear matter density for the model considered in this work. We also find that the part of the quark self-energy that is explicitly dependent upon density has a strong negative Lorentz-scalar term and a strong positive Lorentz-vector term, which is analogous to the self-energy found for the nucleon in nuclear matter when one makes use of the Dirac equation for the nucleon. In this work we calculate the nucleon self -energy in nuclear matter using our model of the quark self-energy and obtain satisfactory results. 
L.S. Celenza, Hu Li, C.M. Shakin, and Qing Sun