Correlations between the nuclear breathing mode energy and properties of asymmetric nuclear matter

Based on microscopic Hartree-Fock + random phase approximation calculations with Skyrme interactions, we study the correlations between the nuclear breathing mode energy $E_{ISGMR}$ and properties of asymmetric nuclear matter with a recently developed analysis method. Our results indicate that the $E_{ISGMR}$ of ${}^{208}$Pb exhibits moderate correlations with the density slope $L$ of the symmetry energy and the isoscalar nucleon effective mass $m_{s,0}^{\ast }$ besides a strong dependence on the incompressibility $K_0$ of symmetric nuclear matter. Using the present empirical values of $L=60\pm 30$ MeV and $m_{s,0}^{\ast }=(0.8\pm 0.1)m$, we obtain a theoretical uncertainty of about $\pm 16$ MeV for the extraction of $K_0$ from the $E_{ISGMR}$ of ${}^{208}$Pb. Furthermore, we find the $E_{ISGMR}$ difference between ${}^{100}$Sn and ${}^{132}$Sn strongly correlates with $L$ and thus provides a potentially useful probe of the symmetry energy.