Effect of high-order empirical parameters on the nuclear equation of state

A quantitative knowledge of the nuclear equation of state (EoS) requires an accurate estimation of the uncertainties on the EoS parameters and their mutual correlations. Such correlations are empirically observed in a large set of EoS models by different authors, but they are not always fully understood. We show that some of these correlations can be interpreted from basic physical constraints imposed on a simple Taylor expansion of the binding energy around saturation density. In particular, we investigate the correlations among the following empirical parameters: the symmetry energy $E_{\mathrm{sym}}$, the slope and curvature of the symmetry energy $L_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$, and the curvature and skewness of the binding energy in symmetric matter $K_{\mathrm{sat}}$ and $Q_{\mathrm{sat}}$. The uncertainties on these correlations is estimated through analytical modeling as well as a metamodeling analysis of the EoS subject to physical constraints. We show that a huge dispersion of the correlations among low-order empirical parameters is induced by the unknown higher-order empirical parameters, such as $K_{\mathrm{sym}}$ (second order) and $Q_{\mathrm{sat}}$ and $Q_{\mathrm{sym}}$ (third order). We also propose an explanation of the reason why $Q_{\mathrm{sat}}$ is weakly constrained by present experimental data. We conclude that selective observables on high-order parameters, such as $K_{\mathrm{sym}}$ and $Q_{\mathrm{sat}}$, should be better determined before the present uncertainties on the EoS can be further reduced.

Figure 1

Expectation values for the parameter ${\alpha }_{EL,i}$ for (a) $i=2$, (b) $i=3$, and (c) $i=4$ as function of $E_{\mathrm{sym}}$ for a set of different type of nuclear interactions considered in Ref. [21]: Skyrme, RMF, and RHF. The points labeled $\chi \mathrm{EFT}$ 2016 stand for the MBPT based on N3LO EFT interaction [28]. The dashed box represents the error on ${\alpha }_{EL,i}$ considering $28<E_{\mathrm{sym}}<36$ MeV [32] and a fixed value for $x_a=-1/9$ (${\alpha }_{EL,i}=-221.5\pm 17.5$ MeV), while the solid box includes additionally the uncertainty $-1/8.5<x_a<-1/9.5$ (${\alpha }_{EL,i}=-220\pm 25\mathrm{MeV}$). See text for more details.

Figure 2

Correlation between the empirical parameters $L_{\mathrm{sym}}$ and $E_{\mathrm{sym}}$ for different kinds of nuclear interactions described in the text (Skyrme, RMF, RHF, and $\chi \mathrm{EFT}$ 2016 [28]) are also plotted. The gray band stands for the uncertainty in the correlation estimated from ${\alpha }_{EL,2}$ (there is almost no difference with the band estimated from ${\alpha }_{EL,4}$). The solid line is the correlation obtained by Farine et al., 1978 [33], Dashed line by Oyamatsu and Iida, 2003 [34], and the ellipses the 68% and 95% confidence intervals of Kortelainen et al., 2010 [35], and dotted band of Holt and Lim, 2018 [38].

Figure 3

Expectations values for ${\alpha }_{K_{\mathrm{sym}},i}$ as function of the linear combination $3E_{\mathrm{sym}}-L_{\mathrm{sym}}$ for a set of different types of energy functionals, as in Fig. 1. The value for $\alpha$ given in Ref. [29] is also shown (purple band).

Figure 4

Correlation between the empirical parameters $K_{\mathrm{sym}}$ and the variable $3E_{\mathrm{sym}}-L_{\mathrm{sym}}$ for different kind of nuclear interactions, as in Fig. 2. The fit from Ref. [29] is shown as well as our analytical expression taken with different order corrections.

Figure 5

Expectations values for ${\alpha }_{KQ,i}$ as function of $K_{\mathrm{sat}}$ for a set of different types of nuclear interactions, as in Fig. 1. The colored band corresponds to ${\alpha }_{KQ,3}$ while the box to ${\alpha }_{KQ,4}$ assuming in addition that $210<K_{\mathrm{sat}}<250$ MeV.

Figure 6

Correlation between the empirical parameters $Q_{\mathrm{sat}}$ and $K_{\mathrm{sat}}$ for different kind of nuclear interactions, as in Fig. 2.

Figure 7

Correlation among the empirical parameters, completed with the effective mass $m_{\mathrm{sat}}^{*}$, the effective mass splitting $\mathrm{\Delta }{m}_{\mathrm{sat}}^{*}$, the parameter $b$, and the variable $3E_{\mathrm{sym}}-L_{\mathrm{sym}}$. The correlation above the diagonal corresponds to the fit of the $\chi \mathrm{EFT}$ [28] predictions only, while below the diagonal, the stability and causality conditions are added up to $0.4{\mathrm{fm}}^{-3}$. See text for discussion.