Toward a Unified Description of Isoscalar Giant Monopole Resonances in a Self-Consistent Quasiparticle-Vibration Coupling Approach

The nuclear incompressibility is a key parameter of the nuclear Equation of State (EoS) that can be extracted from the measurements of the so-called ``breathing mode'' of finite nuclei. The most serious discrepancy so far is between values extracted from Pb and Sn, that has provoked the longstanding question ``Why is tin so soft?". To solve this puzzle, a fully self-consistent Quasiparticle Random-Phase Approximation (QRPA) plus Quasiparticle-Vibration Coupling (QPVC) approach based on Skyrme-Hartree-Fock-Bogoliubov is developed. We show that the many-body correlations introduced by QPVC, which shift the ISGMR energy in Sn isotopes by about 0.4 MeV more than the energy in $^{208}$Pb, play a crucial role in providing a unified description of the ISGMR in Sn and Pb isotopes. The best description of the experimental strength functions is given by SV-K226 and KDE0, which are characterized by incompressibility values $K_\infty=$ 226 MeV and 229 MeV, respectively, at mean field level.