From QCD to nuclear matter saturation

We discuss a relativistic chiral theory of nuclear matter with $\sigma$ and $\omega$ exchange using a formulation of the $\sigma$ model in which all the chiral constraints are automatically fulfilled. We establish a relation between the nuclear response to the scalar field and the QCD one which includes the nucleonic parts. It allows a comparison between nuclear and QCD information. Going beyond the mean field approach we introduce the effects of the pion loops supplemented by the short-range interaction. The corresponding Landau-Migdal parameters are taken from spin-isospin physics results. The parameters linked to the scalar meson exchange are extracted from lattice QCD results. These inputs lead to a reasonable description of the saturation properties, illustrating the link between QCD and nuclear physics. We also derive from the corresponding equation of state the density dependence of the quark condensate and of the QCD susceptibilities