Bayesian Inference of the Symmetry Energy and the Neutron Skin in $^{48}$Ca and $^{208}$Pb from CREX and PREX-2

Using the recent model-independent determination of the charge-weak form factor difference $\Delta F_{\rm CW}$ in $^{48}$Ca and $^{208}$Pb by the CREX and PREX-2 collaborations together with some well-determined properties of doubly magic nuclei, we perform Bayesian inference of the symmetry energy $E_{\rm sym}(\rho)$ and the neutron skin thickness $\Delta r_{\rm np}$ of $^{48}$Ca and $^{208}$Pb within the Skyrme energy density functional (EDF). We find the inferred $E_{\rm sym}(\rho)$ and $\Delta r_{\rm np}$ separately from CREX and PREX-2 are compatible with each other at $90\%$ C.L., although they are inconsistent at $68.3\%$ C.L. with CREX (PREX-2) favoring a very soft (stiff) $E_{\rm sym}(\rho)$ and rather small (large) $\Delta r_{\rm np}$. By combining the CREX and PREX-2 data, we obtain a soft symmetry energy around saturation density $\rho_0$ and thinner $\Delta r_{\rm np}$ of $^{48}$Ca and $^{208}$Pb, which are found to be closer to the corresponding results from CREX alone, implying the PREX-2 is less effective to constrain the $E_{\rm sym}(\rho)$ and $\Delta r_{\rm np}$ due to its lower precision of $\Delta F_{\rm CW}$. Furthermore, we find the Skyrme EDF results inferred by combining the CREX and PREX-2 data nicely agree with the measured dipole polarizabilities $\alpha_D$ in $^{48}$Ca and $^{208}$Pb as well as the neutron matter equation of state from microscopic calculations. The implications of the inferred soft $E_{\rm sym}(\rho)$ around $\rho_0$ are discussed.