_{0}$ are fixed. In Fig. \[XSigmaPara\], we show the $\sigma $ parameter obtained from SHF with MSL0 by varying individually $K_{0}$, $E_{0}(\rho _{0})$, $m_{s,0}^{\ast }$, and $\rho _{0}$. One can see clearly that the $\sigma $ parameter indeed exhibits a strong correlation with $K_{0}$ as expected. However, it also displays a moderate dependence on $m_{s,0}^{\ast }$, a small dependence on $E_{0}(\rho _{0})$, and a very weak correlation with $\rho _{0}$. As can be seen in Fig. \[XSigmaPara\], the $\sigma $ parameter increases with $m_{s,0}^{\ast }$, leading to smaller ISGMR energies according to Eq. (\[EGMRBohigas\]), which is consistent with the results shown in Fig. \[XEaPb208\]. In addition, the fact that $K_{\mathrm{sat,2}}$ parameter decreases with $m_{s,0}^{\ast }$ observed in Fig. \[XKsat2\] will also be partially responsible for the behavior of ISGMR energies decreasing with $m_{s,0}^{\ast }$ as seen in Fig. \[XEaPb208\] since a smaller $K_{\mathrm{sat,2}}$ value will lead to a smaller $E_{\mathrm{ISGMR}} $ as discussed previously.

The above results indicate that the ISGMR centroid energy of $^{208}$Pb exhibits moderate correlations with both $L$ and $m_{s,0}^{\ast }$ besides a strong dependence on $K_{0}$. The accurate knowledge on $L$ and $m_{s,0}^{\ast }$ is thus important for a precise determination of the $K_{0}$ parameter from 