Toward a systematic strategy for defining power counting in the construction of the energy density functional

We propose a new scheme for constructing an effective-field-theory-based interaction to be used in the energy-density-functional (EDF) theory with specific assumptions for defining a power counting. This procedure is developed through the evaluation of the equation of state (EOS) of symmetric and pure neutron matter going beyond the mean-field scheme and using a functional defined up to next-to-leading order (NLO), which we call the NLO EDF. A Skyrme-like interaction is constructed based on the condition of renormalizibility and on power counting on $k_F/{\mathrm{\Lambda }}_{hi}$, where $k_F$ is the Fermi momentum and ${\mathrm{\Lambda }}_{hi}$ is the breakdown scale of our expansion. To absorb the divergences present in beyond mean-field diagrams, counter interactions are introduced for the NLO EDF and determined through renormalization conditions. In particular, three scenarios are explored and all of them lead to satisfactory results. These counter interactions contain also parameters that do not contribute to the EOS of matter and may eventually be determined through future adjustments to properties of some selected finite nuclei. Our work serves as a simple starting point for constructing a well-defined power counting within the EDF framework.

Figure 1

Perturbative expansion of the ground-state energy in a uniform system. The diagrammatic analysis of many-body perturbation theory is for instance illustrated in Ref. [49].

Figure 2

Perturbative expansion of the exact Green's function. We refer the reader to Ref. [49] for details on the diagrammatic representation of the many-body perturbation theory.

Figure 3

The diagrammatic representation of contributions up to NLO for our EDF calculations.

Figure 4

Reference EOS as a function of the density $\rho$ obtained with the SLy5 functional (black dotted line) for symmetric (upper panel) and neutron (lower panel) matter. The LO EOSs (red solid line) are obtained by using Eqs. (2) and (3) with the parameters listed in Table 1.

Figure 5

EOS as a function of the density $\rho$ for symmetric (upper panel) and neutron (lower panel) matter. The dotted line represents the mean-field SLy5 EOSs. The LO parameters listed in Table 1 are used to compute the EOSs generated from the NLO EDF using $V_{\mathrm{LO}}$.

Figure 6

EOS as a function of the density $\rho$, where the subscript SN (NM) represents symmetric (neutron) matter. The dotted (red solid) line represents the mean-field SLy5 (renormalized NLO) EOSs. The NLO EOSs are obtained by using the scenario (abc), that is, Eqs. (16) and (17), with the parameters listed in Table 5.

Figure 7

EOS as a function of the density $\rho$, where the subscript SN (NM) represents symmetric (neutron) matter. The dotted line (blue shaded band) represents the mean-field SLy5 (renormalized NLO) EOSs. The NLO EOSs are obtained by using the scenario (bc), that is, Eqs. (18) and (19), with the parameters listed in Table 6. Here, the cutoff is taken in the window $\mathrm{\Lambda }=1.2$–20 ${\mathrm{fm}}^{-1}$ and the error bars correspond to the cutoff dependence of the fit.

Figure 8

EOSs as a function of the density $\rho$, where the subscript SN (NM) represents symmetric (neutron) matter. The dotted line (green shaded band) represents the mean-field SLy5 (renormalized NLO) EOSs. The NLO EOSs are obtained by using the scenario (c), that is, Eqs. (22) and (23), with the parameters listed in Table 7. Here, the cutoff is taken in the window $\mathrm{\Lambda }=1.2$–20 ${\mathrm{fm}}^{-1}$ and the error bars correspond to the cutoff dependence of the fit.

Figure 9

Second-order EOSs obtained by using Eqs. (18) and (19) [scenario (bc)] at the density value $\rho =0.4{\mathrm{fm}}^{-3}$, as a function of the cutoff $\mathrm{\Lambda }$.

Figure 10

Second-order EOSs obtained by using Eqs. (22) and (23) [scenario (c)] at the density value $\rho =0.4{\mathrm{fm}}^{-3}$, as a function of the cutoff $\mathrm{\Lambda }$.

Figure 11

Parameters in Eqs. (18) and (19) [scenario (bc)] as a function of the cutoff $\mathrm{\Lambda }$. Here the units of parameters are those listed in Table 6.

Figure 12

Parameters in Eqs. (22) and (23) [scenario (c)] as a function of the cutoff $\mathrm{\Lambda }$. Here the units of parameters are those listed in Table 7.