Symmetry conserving non-perturbative s-wave renormalization of the pion in hot and baryonic dense medium

A non-perturbative s-wave renormalization of the pion in a hot and baryon rich medium is presented. This approach proceeds via a mapping of the canonical pion into the axial Noether's charge. The mapping was made dynamical in the Hartree-Fock-Bogoliubov random phase approximation (HFB-RPA). It is shown that this approach, while order mixing, is still symmetry conserving both in the baryon free and baryon rich sectors, at zero as well as finite temperature. The systematic character of this approach is emphasized and it is particularly argued that it may constitute an interesting alternative for the non-perturbative assessment of the nuclear matter saturation properties.