Constraining the density dependence of the symmetry energy with nuclear data and astronomical observations in the Korea-IBS-Daegu-SKKU framework

Background: The properties of very neutron rich nuclear systems are largely determined by the density dependence of the nuclear symmetry energy. The Korea-IBS-Daegu-SKKU (KIDS) framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore the symmetry-energy parameters such as $J$ (value at saturation density), $L$ (slope at saturation), $K_{\mathrm{sym}}$ (curvature at saturation), and higher-order ones independently of each other and independently of assumptions about the in-medium effective mass, as previously shown in the cases of closed-shell nuclei and neutron-star properties.

Purpose: We examine the performance of EoSs with different symmetry energy parameters on properties of nuclei and observations of neutron stars and gravitational waves in an effort to constrain in particular $L$ and $K_{\mathrm{sym}}$ or the droplet-model counterpart $K_{\tau }$.

Method: Assuming a standard EoS for symmetric nuclear matter, we explore several points on the hyperplane of $(J,L,K_{\mathrm{sym}}$ or $K_{\tau })$ values. For each point, the corresponding KIDS functional parameters and a pairing parameter are obtained for applications in spherical even-even nuclei. This is the first application of KIDS energy density functionals with pairing correlations in a spherical Hartree-Fock-Bogoliubov computational code. The different EoSs are tested successively on the properties of closed-shell nuclei, along the Sn isotopic chain, and on astronomical observations, in a step-by-step procedure of elimination and correction.

Results: A small regime of best-performing parameters is determined and correlations between symmetry-energy parameters are critically discussed. The results strongly suggest that $K_{\mathrm{sym}}$ is negative and no lower than $-200\mathrm{MeV}$, that $K_{\tau }$ lies between roughly $-400$ and $-300\mathrm{MeV}$ and that $L$ lies between 40 and 65 MeV, with $L⪅55\mathrm{MeV}$ more likely. For the selected well-performing sets, corresponding predictions for the position of the neutron drip line and the neutron skin thickness of selected nuclei are reported. The results are only weakly affected by the choice of effective mass values. Parts of the drip line can be sensitive to the symmetry energy parameters.

Conclusion: Using KIDS EoSs for unpolarized homogeneous matter at zero temperature and KIDS EDFs with pairing correlations in spherical symmetry we have explored the hyperplane of symmetry-energy parameters. Using both nuclear-structure data and astronomical observations as a testing ground, a narrow regime of well-performing parameters has been determined, free of nonphysical correlations and decoupled from constraints on the nucleon effective mass. The results underscore the role of $K_{\tau }$ and of precise astronomical observations. More-precise constraints are possible with precise fits to nuclear energies and, in the future, more-precise input from astronomical observations.

Figure 1

Dependence of results on kinetic terms (effective mass) illustrated for two EoSs. (a)–(c) KIDS-P4 EoS, $(J,L,K_{\mathrm{sym}}/K_{\tau },Q_{\mathrm{sym}})=(33,49,-157/-374,583)\mathrm{MeV}$ (rounded values); (d)–(f) $(J,L,K_{\mathrm{sym}}/K_{\tau },Q_{\mathrm{sym}})=(30,50,-78/-300,650)\mathrm{MeV}$. (a), (d) Binding energy per particle $E_B/A$ of the indicated closed-shell nuclei; lines are drawn to guide the eye. (b), (e) Two-neutron separation energy $S_{2n}$ of Ca isotopes. (c), (f) $S_{2n}$ of Sn isotopes. The assumed effective mass values $({\mu }_s,{\mu }_v)$ are indicated or the prescription $y_1=y_2=0$.

Figure 2

Average deviation per datum (ADPD) for the indicated $J$ values (a) as a function of $K_{\tau }$ and (b) as a function of $L$. The various points at given $(J,K_{\tau })$ correspond to different $L$ values, while the various points at given $(J,L)$ to different $K_{\tau }$ values.

Figure 3

Fitting quality on 13 data as a function of $L$ considering the shown values of $J,K_{\tau }$ and $({\mu }_s,{\mu }_v)=(0.9,0.9)$.

Figure 4

Mass-radius relation with symmetry energy parameters ${(J,L,K_{\mathrm{sym}}/K_{\tau })}_{\mathrm{CS}}$ noted in the panel in units of MeV. Observational data include large masses in horizontal bands, x-ray burst analysis in gray bands, and two bars with errors from NICER observation.

Figure 5

Particle fraction in the core of neutron stars with the indicated ${(J,L,K_{\mathrm{sym}}/K_{\tau })}_{\mathrm{CS}}$ values in MeV.

Figure 6

Same as Fig. 4, but for the “$\mathrm{CS}+\mathrm{OS}$” parameter sets.

Figure 7

Constraints on and correlations between (a) $K_{\mathrm{sym}}$ and $L$ and (b) $K_{\mathrm{sym}}$ and $(3J-L)$. The filled black points (squares, disks, triangles) correspond to our tentatively selected KIDS EoSs and they are connected according to the constraints they obey (CS: closed-shell nuclei; OS: open-shell nuclei; NS: neutron stars—see text). In panel (a) the points are labeled also by the $K_{\tau }$ value in MeV. Filled points inside magenta circles correspond to the EoSs providing the best ADPD along the Sn chain, while crosses represent EoSs which were discarded in Secs. 3a3 and 3b. The open circle corresponds to the KIDS-P4 EoS [11]. The inverted triangle corresponds to the newly examined point (see text) with the corresponding ($J,L,K_{\mathrm{sym}}/K_{\tau }$) values given in MeV. Ellipses circumscribe regimes free of rejected points. Lines depict correlations found in the literature (Tews et al. [45], Holt et al. [46], Mondal et al. [20]) except the line labeled “approx.” which corresponds to Eq. (11). The point with error bars labeled “Tsang et al.” represents the results reported in Ref. [8].