Electric Dipole Polarizability in $^{208}$Pb as a Probe of the Symmetry Energy and Neutron Matter around $\rho_0/3$

It is currently a big challenge to accurately determine the symmetry energy $E_{\text{sym}}(\rho)$ and the pure neutron matter equation of state $E_{\text{PNM}}(\rho)$, even their values around saturation density $\rho_0 $. We find that the electric dipole polarizability $\alpha _ {\text{D}}$ in $^{208}$Pb can be determined uniquely by the magnitude of the $E_{\text{sym}}(\rho)$ or almost equivalently the $E_{\text{PNM}}(\rho)$ at subsaturation densities around $\rho_0/3 $, shedding a light upon the genuine correlation between the $\alpha _ {\text{D}}$ and the $E_{\text{sym}}(\rho)$. By analyzing the experimental data of the $\alpha _ {\text{D}}$ in $^{208}$Pb from RCNP using a number of non-relativistic and relativistic mean-field models, we obtain very stringent constraints on $E_{\text{sym}}(\rho)$ and $E_{\text{PNM}}(\rho)$ around $\rho_0/3 $. The obtained constraints are found to be in good agreement with the results extracted in other analyses. In particular, our results provide for the first time the experimental constraints on $E_{\text{PNM}}(\rho)$ around $\rho_0/3 $, which are in harmony with the recent determination of $E_{\text{PNM}}(\rho)$ from microscopic theoretical studies and potentially useful in constraining the largely uncertain many-nucleon interactions in microscopic calculations of neutron matter.