Density dependence of the nuclear energy-density functional

Background: The explicit density dependence in the coupling coefficients entering the nonrelativistic nuclear energy-density functional (EDF) is understood to encode effects of three-nucleon forces and dynamical correlations. The necessity for the density-dependent coupling coefficients to assume the form of a preferably small fractional power of the density $\rho$ is empirical and the power is often chosen arbitrarily. Consequently, precision-oriented parametrizations risk overfitting in the regime of saturation and extrapolations in dilute or dense matter may lose predictive power.

Purpose: Beginning with the observation that the Fermi momentum $k_F$, i.e., the cubic root of the density, is a key variable in the description of Fermi systems, we first wish to examine if a power hierarchy in a $k_F$ expansion can be inferred from the properties of homogeneous matter in a domain of densities, which is relevant for nuclear structure and neutron stars. For subsequent applications we want to determine a functional that is of good quality but not overtrained.

Method: For the EDF, we fit systematically polynomial and other functions of ${\rho }^{1/3}$ to existing microscopic, variational calculations of the energy of symmetric and pure neutron matter (pseudodata) and analyze the behavior of the fits. We select a form and a set of parameters, which we found robust, and examine the parameters' naturalness and the quality of resulting extrapolations.

Results: A statistical analysis confirms that low-order terms such as ${\rho }^{1/3}$ and ${\rho }^{2/3}$ are the most relevant ones in the nuclear EDF beyond lowest order. It also hints at a different power hierarchy for symmetric vs. pure neutron matter, supporting the need for more than one density-dependent term in nonrelativistic EDFs. The functional we propose easily accommodates known or adopted properties of nuclear matter near saturation. More importantly, upon extrapolation to dilute or asymmetric matter, it reproduces a range of existing microscopic results, to which it has not been fitted. It also predicts a neutron-star mass-radius relation consistent with observations. The coefficients display naturalness.

Conclusions: Having been already determined for homogeneous matter, a functional of the present form can be mapped onto extended Skyrme-type functionals in a straightforward manner, as we outline here, for applications to finite nuclei. At the same time, the statistical analysis can be extended to higher orders and for different microscopic (ab initio) calculations with sufficient pseudodata points and for polarized matter.

Figure 1

Representative results ($\beta =1/2$) of fits with two, three, or four parameters, as indicated, along with the APR pseudodata, the DSS results from chiral EFT and the SLy4 functional. Shown is the potential energy per particle divided by the density as a function of ${\rho }^{1/3}$, in SNM and PNM.

Figure 2

Results of various fits are shown, for the energy per particle in SNM, PNM, and asymmetric matter, along with the APR pseudodata, the APR parametrization of the low-density phase (LDP), the DSS results from chiral EFT and the SLy4 functional. The fits of the terms $c_0,c_1,c_2,c_3$ with $\beta =0,1/2,1$ are included, as well as the parametrization with SNM parameters adapted to the saturation point, labeled “ad-1” in Table 3. Results with the “ad-2” set would be almost indistinguishable on the figure.

Figure 3

Mass and radius relation of neutron stars for the models of Table 3. The two horizontal bands represent the minimum of maximum mass of neutron stars [3, 4]. The central region shows the allowed area of mass and radius of neutron stars analyzed from x-ray burst data [49]. The corresponding values for the central baryon density and the maximum mass $({\varrho }_c,M)$ in units of (${\mathrm{fm}}^{-3},M_{\odot })$ are (1.135, 2.20) ($\beta =0$), (1.140, 2.18) ($\beta =1/2$), (1.165, 2.10) ($\beta =1$), (1.135, 2.14) ($\beta =1/2$, adapted to the saturation point of SNM).