Application to the Gogny interaction

In this section, we apply the results of Sec. 3 to the finite-range part of the Gogny interaction D1S [17]. This amounts to calculating moments (52) and (53) of the Gaussian functions with the two ranges of 0.7 and 1.2fm, which constitute the central part of the Gogny interaction. Because this interaction does not contain any finite-range spin-orbit or tensor force in Eqs. (50) and (51), the moments $P$, $Q$, $R$, and $S$ are set to zero. On the other hand, the zero-range spin-orbit and density-dependent terms of the Gogny interaction are left unchanged.

Table 1: The NV coupling constants (49)-(50) calculated for the Gogny interaction D1S [17] for the Fermi momenta of $k_F=0$ and 1.35fm$^{-1}$. First-order coupling constants, $C_t^{\rho }$ and $C_t^{\Delta s}$, are in units of MeVfm$^3$ and the second-order coupling constants, $C_t^{\Delta \rho }$, $C_t^{\tau }$, $C_t^{\Delta s}$, and $C_t^{T}$, are in units of MeVfm$^5$.

	$t=0$		$t=1$
	$k_F=0$	$k_F=1.35$	$k_F=0$	$k_F=1.35$
$C_t^{\rho }$	$-$665.1658	$-$600.6156	468.5360	428.3580
$C_t^{s}$	$-$25.09219	$-$57.13246	221.7219	230.3318
$C_t^{\Delta \rho }$	$-$125.3365	$-$84.66327	56.65570	31.22042
$C_t^{\tau }$	236.9227	74.22964	$-$141.9368	$-$40.19573
$C_t^{\Delta s}$	10.72944	$-$10.25276	58.80425	65.14281
$C_t^{T}$	$-$80.70182	3.226982	$-$10.87263	$-$36.22687

Table 2: The standard Skyrme-force parameters (57)-(59) calculated for the Gogny interaction D1S [17] for the Fermi momenta of $k_F=0$ and 1.35fm$^{-1}$. These parameters correspond to the time-even sector of the Skyrme functional. Parameter $t_0$ is in units of MeVfm$^3$; parameters $t_1$ and $t_2$ are in units of MeVfm$^5$, and parameters $x_0$, $x_1$, and $x_2$ are dimensionless.

	$k_F=0$	$k_F=1.35$
$t_0$	$-$1773.775	$-$1601.642
$t_1$	984.3584	550.5103
$t_2$	810.3964	$-$166.0710
$x_0$	0.5565848	0.5697973
$x_1$	0.2488322	0.09972511
$x_2$	$-$0.9915809	$-$0.5517191

In Table 1 we show values of coupling constants (49)-(51) calculated in the vacuum ($k_F=0$) and at the saturation density ($k_F=1.35$fm$^{-1}$). Similarly, Table 2 shows values of the Skyrme-force parameters (57)-(59) corresponding to the time-even coupling constants. In Figs. 2 and 3 we plot the coupling constants and Skyrme-force parameters, respectively, as functions of the Fermi momentum.

Figure 2: Dependence of the NV coupling constants (49)-(50), calculated for the Gogny interaction D1S [17], on the Fermi momentum $k_F$. Full and open symbols (lower and upper panels) show values of the time-even and time-odd coupling constants, respectively. Solid and dashed lines (left and right panels) show values of the isoscalar and isovector coupling constants, respectively. Units are specified in the caption to Table 1.

Figure 3: Dependence of the standard Skyrme-force parameters (57)-(59), calculated for the Gogny interaction D1S [17], on the Fermi momentum $k_F$. These parameters correspond to the time-even sector of the Skyrme functional. Units are specified in the caption of Table 2.

The most important observation resulting from values shown in Tables 1 and 2 and Figs. 2 and 3 pertains to a significant density (or $k_F$) dependence of the coupling constants and Skyrme-force parameters. The strongest dependence is obtained for the isoscalar tensor coupling constant $C_0^{T}$. (Note that the central finite-range Gogny interaction induces significant values of the tensor coupling constants $C_t^{T}$, even if this force does not contain any explicit tensor term.) Also the kinetic coupling constants $C_t^{\tau }$ exhibit a strong density dependence, going almost to zero at $k_F\sim2$fm$^{-1}$. We note that the obtained density dependencies do not, in general, follow any power laws. Significantly stronger density dependencies are obtained for the Skyrme-force parameters (Fig. 3). The pole appearing in the parameter $x_2$ is a consequence of the fact that parameters $t_2$ and $t_2x_2$, derived in Eq. (59), change signs at slightly different values of the Fermi momentum.

Figure 4: Dependence of the infinite-matter energy per particle on the density (left panels) or Fermi momentum (right panels). Thick lines show the NV results for density-dependent coupling constants. Thin lines show the NV results for fixed coupling constants calculated at the given values of $k_F^0$. Upper and lower panels show results for fixed coupling constants (67) and for coupling constants (68) that depend on density as $\rho ^{2/3}$. The full square and circles mark the minima of curves.

Let us now consider the question of whether parameters calculated at any fixed value of the Fermi momentum can provide a reasonable alternative. To analyze this point, we first note that the zero-order coupling constants given in Eq. (49) depend on $k_F$ (i) implicitly through the moments of Eq. (53) and (ii) explicitly through the $k_F^2$ term, that is,

$$C_t^{\rho}(k_F) = C_{t,0}^{\rho}(k_F) + C_{t,2/3}^{\rho}(k_F)\rho^{2/3} ,$$ (65)

$$C_t^{s }(k_F) = C_{t,0}^{s }(k_F) + C_{t,2/3}^{s }(k_F)\rho^{2/3} ,$$ (66)

where we employed the standard association of the Fermi energy with density, namely, $\rho=2k_F^3/3\pi^2$. Therefore, we can consider fixed values of the coupling constants calculated at a given Fermi momentum $k_F^0$ as

$$C_t^{\rho}(k_F^0) \mbox{~and~} C_t^{s }(k_F^0), \mbox{~or~}$$ (67)

$$C_{t,0}^{\rho}(k_F^0), C_{t,2/3}^{\rho}(k_F^0), C_{t,0}^{s }(k_F^0), \mbox{~and~} C_{t,2/3}^{s }(k_F^0).$$ (68)

The second option gives the coupling constants that still depend on the density as $\rho ^{2/3}$, in analogy with the standard density-dependent term, which for the Gogny force depends on the density as $\rho^{1/3}$, and which in the present study is alway kept untouched.

In the upper and lower panels of Fig. 4, thin lines show the nuclear matter equations of state (energy per particle in function of density or Fermi momentum) obtained for the coupling constants fixed according to prescriptions (67) and (68), respectively. For comparison, thick lines show the NV results obtained for density-dependent coupling constants. Since in the nuclear matter, the factor multiplying the term $\nu_0({r})\nu_2({r})$ in Eq. (18) vanishes exactly (by construction), the thick lines correspond to the exact Gogny force results.

Equations of state calculated with prescription (68) for fixed values of $k_F^0$ from 1.35 to 1.98fm$^{-1}$ with the step of 0.07fm$^{-1}$ completely miss the saturation point. This means that Skyrme forces with constant parameters (68) derived by the NV expansion cannot be equivalent to the finite-range Gogny interaction. On the other hand, prescription (67), for fixed value of $k_F^0=1.35$fm$^{-1}$, reproduces the equation of state fairly well, with some deviations seen only at densities beyond the saturation point. For comparison, we also show results obtained with $k_F^0=1$ and 1.7fm$^{-1}$, which fit the equation of state at low and high densities, respectively. These results show that the explicit dependence on the Fermi momentum, which appears in the NV expansion, cannot be used to define the density dependence of the coupling constants.

Figure 5: Similar to Fig. 2 except for the time-odd NV coupling constants only. The open symbols show exact results (49)-(50) as plotted in Fig. 2, whereas the full symbols show results inferred from the time-even sector by using Eqs. (62) and (63). Solid and dashed lines (left and right panels) show values of the isoscalar and isovector coupling constants, respectively.

Table 3: Similar to Table 1 except for the time-odd NV coupling constants inferred from the time-even sector by using Eqs. (62) and (63) for a Fermi momentum of 1.35fm$^{-1}$.

	$t=0$	$t=1$
$C_t^{s}$	$-$27.94756	200.2052
$C_t^{\Delta s}$	20.92673	23.21032
$C_t^{T}$	$-$26.47083	$-$44.78631

The full density dependence of all Skyrme-force parameters was recently implemented in a spherical self-consistent code [31]. Here we use this implementation to test the results of the NV expansion against the full-fledged solutions known for the Gogny D1S force [32,33]. In Table 4, we show results obtained for three sets of Skyrme-force parameters:

• S1Sa: fixed parameters given in Table 2 for $k_F^0=1.35$fm$^{-1}$, used together with the standard D1S parameters [17] of the zero-range spin-orbit ($W=130$MeVfm$^5$) and density-dependent ($t_3=1390.60$MeVfm$^4$) terms.
• S1Sb: density-dependent parameters shown in Fig. 3, implemented for $\rho=2k_F^3/3\pi^2$ and used with the standard values of $W$ and $t_3$.
• S1Sc: fixed parameters identical to those of S1Sa, but with the value of $t_3=1385.35$MeVfm$^4$ slightly changed, so as to bring the ground-state energy of $^{208}$Pb exactly to the true D1S value [33].

Table 4 shows the ground-state energies $E$ of seven doubly magic spherical nuclei, calculated by using the Skyrme-force parameters S1Sa, S1Sb, and S1Sc, and compared with the Gogny-force energies $E_G$. To facilitate the comparison, we also show relative differences $\Delta{E}=(E-E_G)/\vert E_G\vert$ in percent along with the corresponding RMS deviations (the last row in Table 4).

On can see that for the Skyrme-force parameters S1Sa and S1Sb, which are directly derived from the Gogny force by using the NV expansion, one obtains the nuclear binding energies smaller by 1-2% as compared to those given by the original Gogny force. This should be considered a very good result, although it cannot compete in precision of describing experimental data with the original Gogny or Skyrme forces, which have parameters directly fitted to experimental binding energies. A simple rescaling of the parameter $t_3$ brings the RMS deviation to below $0.4\%$, and makes the Skyrme force S1Sc competitive with most other standard Skyrme parameterizations. At the same time, the analogous RMS deviations obtained for the neutron and proton radii are 0.20 and 0.30% (S1Sa), 1.01 and 0.94% (S1Sb), and 0.26 and 0.44% (S1Sc), respectively.

We note here in passing that in this study we defined the S1Sb parameter set by considering the density-dependent coupling constants $C_t^{\Delta \rho }$ (Fig. 3) that multiply densities $\rho_t\Delta\rho_t$. An attempt of using the same coupling constants along with densities $-(\bbox{\nabla}\rho_t)^2$ gives, in fact, the RMS deviations of binding energies (5.34%), neutron radii (2.44%), and proton radii (2.27%), which are significantly worse than those of S1Sb (Table 4). This shows that the prescription to replace in EDFs integrated by parts (see Refs. [29,31]) the Fermi momentum by $\rho=2k_F^3/3\pi^2$ may lead to significantly different results. Another recipy is to associate $k_F$ with density before taking products of the two density matrices which means both of the above terms will be active. However in order to have a better correspondence with the traditional Skyrme functionals we have made this association after taking the product.

It is not our purpose here to propose that any of the Skyrme-force parameterizations introduced in the present work are better solutions to the problem of finding the best agreement with data. It is already known that within the standard second-order Skyrme-force parameterizations, a spectroscopic-quality [2] force cannot be found [34]. Nevertheless, it is gratifying to see that the NV expansion allows us to bridge the gap between the non-local and quasi-local EDFs, or between the finite-range and zero-range effective forces. A more quantitative discussion of the accuracy of the NV expansion will be possible by considering higher-order NV expansions [35].

Table 4: Binding energies $E$ of seven doubly magic nuclei calculated by using the Skyrme-force parameters S1Sa, S1Sb, and S1Sc (see text) compared with the Gogny-force energies $E_G$. All energies are in MeV.

	D1S [33]	S1Sa		S1Sb		S1Sc
	$E_G$	$E$	$\Delta{E}$	$E$	$\Delta{E}$	$E$	$\Delta{E}$
$^{ 40}$Ca	$-$342.689	$-$335.312	2.15%	$-$340.642	0.60%	$-$339.369	0.97%
$^{ 48}$Ca	$-$414.330	$-$409.118	1.26%	$-$410.698	0.88%	$-$414.213	0.03%
$^{ 56}$Ni	$-$481.111	$-$473.497	1.58%	$-$471.970	1.90%	$-$479.843	0.26%
$^{ 78}$Ni	$-$637.845	$-$630.447	1.16%	$-$629.066	1.38%	$-$638.837	$-$0.16%
$^{100}$Sn	$-$828.024	$-$814.568	1.63%	$-$814.896	1.59%	$-$826.453	0.19%
$^{132}$Sn	$-$1101.670	$-$1086.272	1.40%	$-$1086.867	1.34%	$-$1101.445	0.02%
$^{208}$Pb	$-$1637.291	$-$1612.634	1.51%	$-$1617.419	1.21%	$-$1637.291	0.00%
RMS	n.a.	n.a.	1.56%	n.a.	1.33%	n.a.	0.39%

Finally, in Fig. 5 we compare the time-odd coupling constants calculated by using Eqs. (49) and (50) with those corresponding to the Skyrme-force parameters; that is, calculated by using Eqs. (62) and (63). Similarly, Table 3 lists the numerical values of the Skyrme-force time-odd coupling constants. As one can see, differences between both sets of the time-odd coupling constants, shown in Fig. 5 with open and full symbols, are quite substantial. These results illustrate the fact that the NV expansion of the Gogny force leads to the Skyrme functional and not to the Skyrme force.

We conclude this section by noting that functions $\pi _i(r)$ approximated by Gaussians (see Eqs. (36)-(38) and Fig. 1) lead to the coupling constants and Skyrme-force parameters, which, when plotted in the scales of Figs. 2, 3, and 5, are indistinguishable from those presented in these figures.