Electric dipole polarizability in ${}^{208}\mathbf{Pb}$ 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 $E_{\mathrm{sym}}\left(\rho \right)$ and the pure neutron matter equation of state $E_{\mathrm{PNM}}\left(\rho \right)$ even their values around saturation density ${\rho }_0$. We find that the electric dipole polarizability ${\alpha }_{\mathrm{D}}$ in ${}^{208}\mathrm{Pb}$ can be determined uniquely by the magnitude of $E_{\mathrm{sym}}\left(\rho \right)$ or almost equivalently $E_{\mathrm{PNM}}\left(\rho \right)$ at subsaturation densities around ${\rho }_0/3$, shedding light upon the genuine correlation between ${\alpha }_{\mathrm{D}}$ and $E_{\mathrm{sym}}\left(\rho \right)$. By analyzing the experimental data of ${\alpha }_{\mathrm{D}}$ in ${}^{208}\mathrm{Pb}$ from the Research Center for Nuclear Physics using a number of nonrelativistic and relativistic mean-field models, we obtain very stringent constraints on $E_{\mathrm{sym}}\left(\rho \right)$ and $E_{\mathrm{PNM}}\left(\rho \right)$ 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 $E_{\mathrm{PNM}}\left(\rho \right)$ around ${\rho }_0/3$, which are in harmony with the recent determination of $E_{\mathrm{PNM}}\left(\rho \right)$ from microscopic theoretical studies and potentially useful in constraining the largely uncertain many-nucleon interactions in microscopic calculations of neutron matter.

Figure 1

${10}^3/{\alpha }_{\mathrm{D}}$ in ${}^{208}\mathrm{Pb}$ vs $E_{\mathrm{sym}}\left({\rho }_r\right)$ at (a) ${\rho }_r=0.02{\mathrm{fm}}^{-3}$, (b) $0.05{\mathrm{fm}}^{-3}$, (c) $0.08{\mathrm{fm}}^{-3}$, (d) $0.11{\mathrm{fm}}^{-3}$, and (e) $0.16{\mathrm{fm}}^{-3}$, predicted by a large number (62) of nonrelativistic (SHF) and RMF models. The shaded regions correspond to linear fitting of ${10}^3/{\alpha }_{\mathrm{D}}=a+b{E}_{\mathrm{sym}}\left({\rho }_r\right)$ with $1\sigma$ and $2\sigma$ confidence bands, and $r$ is the Pearson correlation coefficient. The red hatched band corresponds to the experimental result of ${\alpha }_{\mathrm{D}}=20.1\pm 0.6{\mathrm{fm}}^3$ from RCNP [33].

Figure 2

Constraints on the symmetry energy $E_{\mathrm{sym}}\left(\rho \right)$ as a function of density $\rho$ (see text for the details). The inset shows the density dependence of the Pearson correlation coefficient $r$ between $1/{\alpha }_{\mathrm{D}}$ in ${}^{208}\mathrm{Pb}$ and $E_{\mathrm{sym}}\left(\rho \right)$.

Figure 3

Constraints and predictions for the pure neutron matter EOS $E_{\mathrm{PNM}}\left(\rho \right)$ as a function of density $\rho$ (see text for the details). The inset shows the Pearson correlation coefficient $r$ between $1/{\alpha }_{\mathrm{D}}$ in ${}^{208}\mathrm{Pb}$ and $E_{\mathrm{PNM}}\left(\rho \right)$ as a function of $\rho$.