Constraints on the nuclear symmetry energy from asymmetric-matter calculations with chiral $NN$ and $3N$ interactions

The nuclear symmetry energy is a key quantity in nuclear (astro)physics. It describes the isospin dependence of the nuclear equation of state, which is commonly assumed to be almost quadratic. In this work, we confront this standard quadratic expansion of the equation of state with explicit asymmetric nuclear-matter calculations based on a set of commonly used Hamiltonians including two- and three-nucleon forces derived from chiral effective-field theory. We study, in particular, the importance of nonquadratic contributions to the symmetry energy, including the nonanalytic logarithmic term introduced by Kaiser [Phys. Rev. C 91, 065201 (2015)]. Our results suggest that the nonquadratic contribution to the symmetry energy can be systematically determined from the various Hamiltonians employed, and we obtain $0.{74}_{-0.08}^{+0.11}$ MeV (or $-1.{02}_{-0.08}^{+0.11}$ MeV for the potential term with the effective-mass contribution) at nuclear saturation density, while the logarithmic contribution to the symmetry energy is relatively small and model-dependent. We also employ the meta-model approach to study the impact of the higher-order contributions on the neutron-star crust-core transition density, and find a 5% correction.

Figure 1

Comparison of the MBPT predictions for the (a) energy per particle in PNM [27] (blue points) with the APR EOS [51] (green squares), QMC calculations Wlazlowski et al. (2014) [31] (cyan triangles), and Tews et al. (2016) [52] (black dots) using different chiral EFT Hamiltonians with $NN$ and $3N$ forces. The latter points include simple estimates for the EFT truncation error of the chiral expansion. We also show our fit posterior at 68% (95%) confidence level as dark (light) red bands. (b) The same comparison but with a different scaling.

Figure 2

Results of the Bayesian parametric fits of the Landau mass in (a) SNM and (b) PNM. The 68% (95%) confidence levels for the posterior distribution functions are shown as dark-shaded (light-shaded) bands. The black lines represent the individual Hamiltonians, and the black points show the average over the six Hamiltonians with $\pm 1\sigma$ uncertainty bands.

Figure 3

Comparison of the Bayesian inference results for the MM of this work (red bands) with the MBPT data (blue points) for SNM [panels (a)–(c)] and PNM [panels (d)–(f)] and the three different scalings described in the main text. The bands are given at the 65% (dark-shaded) and 95% confidence level (light-shaded), whereas the data points are shown with the $\pm 1\sigma$ uncertainty estimate.

Figure 4

Results for the (a) symmetry energy, $e_{\text{sym}}\left(n\right)$, (b) its potential contribution $e_{\text{sym}}^{\text{pot}}$, and (c) the effective potential $e_{\text{sym}}^{\text{pot}*}$ using (21) and (22). The meaning of the individual bands and points is the same as in Fig. 3. In the right panel, the light- and dark-shaded red bands and the data (blue points) correspond to calculations where the Landau mass is represented by a linear polynomial. The black squares (without error bars) and the black dashed lines (enclosing a band) represent calculations where the Landau mass is represented by a quadratic fit.

Figure 5

Comparison of the extracted $e_{\text{sym},2}$, $e_{\text{sym},2}^{\text{pot}}$, and $e_{\text{sym},2}^{\text{pot}*}$ using Eqs. (24, 25, 26) [panels (a)–(c)] and via an expansion around PNM, i.e., using Eq. (31) [panels (d)–(f)]. Panels (g)–(i) show the difference between the $\delta$ and $\eta$ expansions.

Figure 6

Nonquadratic terms (a) $e_{\text{sym},\mathrm{nq}}$, (b) $e_{\text{sym},\mathrm{nq}}^{\mathrm{pot}}$, and (c) $e_{\text{sym},\mathrm{nq}}^{\mathrm{pot}*}$ calculated via an expansion around SNM (blue) and PNM (red). For panel (c), the Landau mass is described by a linear fit. The colored lines depict results for the six individual Hamiltonians.

Figure 7

Residuals $R$ of the model, see Eq. (33), with respect to the data as a function of the asymmetry parameter $\delta$ for different values of the density: $n=0.06{\mathrm{fm}}^{-3}$ [panels (a)–(c)], $n=0.12{\mathrm{fm}}^{-3}$ [panels (d)–(f)], and $n=0.16{\mathrm{fm}}^{-3}$ [panels (g)–(i)]. The results are shown for the two different calculations of $y_{\mathrm{sym},2}$ and $y_{\mathrm{sym},\mathrm{nq}}$—the expansions around PNM (red) and SNM (blue). The different columns correspond to $e$, $e^{\text{pot}}$, and $e^{\text{pot},*}$. The colored lines depict the residuals of the fit for each Hamiltonian. In the last column, the black-dashed lines represent the upper and lower limits of the uncertainty in the residuals, respectively, by disregarding the uncertainties in the effective masses.

Figure 8

Predictions for the spinodal density obtained by solving Eq. (34), and for beta equilibrium in low-density uniform matter by solving Eq. (35). The intersection denotes the crust-core transition, as indicated by a dot in the inset. The quadratic approximation (red band) is compared with the case where quartic contributions are included (blue band).

Figure 9

The neutron single-particle energies ${\epsilon }_n\left(k\right)$ as a function of the momentum $k$ calculated at $n_{\text{sat}}^{\mathrm{emp}}$, and extracted from the MBPT calculations of Ref. [27]. The different colors correspond to the six Hamiltonians as labeled in the legend. We show the single-particle energies obtained from (a) only $NN$ forces and (b) when including $3N$ contributions. In each panel, we present results for both SNM and PNM.

Figure 10

Same as Fig. 9 but for the neutron effective mass as function of the momentum $k$.

Figure 11

(a) Landau effective mass and (b) its inverse in SNM and PNM as a function of density. The black-dashed lines represent the upper and lower limits when only $NN$ forces are considered, while the gray-shaded regions show the results with $3N$ forces included. The different colors correspond to the six Hamiltonians as labeled in the legend.