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Determination of symmetry energy from experimental and observational constraints; prediction on CREX

Taking $r_{\rm skin}^{208}({\rm PREX})=0.33^{+0.16}_{-0.18}~{\rm fm}$ as an experimental constraint and $M_{\rm max}^{\rm NS} \ge 2{\rm M}_{\rm sun}$ as an observational (astrophysical) constraint, we determine an indisputable range for $J$, $L$, $K_{\rm sym}$ defined in Eq.~\eqref{eq-S-sym}. For this purpose, we take a statistical approach. We first accumulate the 206 EoS data from theoretical works and take correlation between $r_{\rm skin}^{208}$ and $L$ for the 206 EoSs, where 7 Gogny EoSs are obtained by our calculations. Since the correlation coefficient is $R = 0.99$, we can regard $L$ as a function of $r_{\rm skin}^{208}$, so that we succeed in deducing an empirical constraint $L=31-161$~MeV from $r_{\rm skin}^{208}({\rm PREX})=0.15-0.49$~fm. For the 47 EoSs satisfying the observational constraint, 46 EoSs satisfy the empirical constraint. The 46 EoSs yield $J=29-44$~MeV, $L= 37-135$~MeV, $K_{\rm sym}=(-137)-(160)$~MeV. The is a primary result. When we take correlation between $r_{\rm skin}^{48}$ and $r_{\rm skin}^{208}$ for the 206 EoSs, $R$ is 0.99. The $r_{\rm skin}^{48}$--$r_{\rm skin}^{208}$ relation allows us to transform $r_{\rm skin}^{208}({\rm PREX})$ into the corresponding data on $r_{\rm skin}^{48}$ In order to estimate a value of $r_{\rm skin}^{48}$ for ongoing CREX, we take the weighted mean and its error of two present data on $r_{\rm skin}^{48}$ and transformed PREX value on $r_{\rm skin}^{48}$. The weighted mean is $r_{\rm skin}^{48}=0.17$~fm. This is a prediction for the central value of CREX.