Propagation of statistical uncertainties in covariant density functional theory: Ground state observables and single-particle properties

Statistical errors in ground state observables and single-particle properties of spherical even-even nuclei and their propagation to the limits of nuclear landscape have been investigated in covariant density functional theory (CDFT) for the first time. In this study we consider only covariant energy density functionals with nonlinear density dependency. Statistical errors for binding energies and neutron skins significantly increase on approaching the two-neutron drip line. On the contrary, such a trend does not exist for statistical errors in charge radii and two-neutron separation energies. The absolute and relative energies of the single-particle states in the vicinity of the Fermi level are characterized by low statistical errors [$\sigma \left(e_i\right)\sim 0.1$ MeV]. Statistical errors in the predictions of spin-orbit splittings are rather small. Statistical errors in physical observables are substantially smaller than related systematic uncertainties. Thus, at the present level of the development of theory, theoretical uncertainties at nuclear limits are dominated by systematic ones. Statistical errors in the description of physical observables related to the ground state and single-particle degrees of freedom are typically substantially lower in CDFT as compared with Skyrme density functional theory. The correlations between the model parameters are studied in detail. The parametric correlations are especially pronounced for the $g_2$ and $g_3$ parameters which are responsible for the density dependence of the model. The accounting of this fact potentially allows us to reduce the number of free parameters of the nonlinear meson coupling model from 6 to 5.

Figure 1

The range of the variations of the parameters of the NL5(C) CEDF and related changes of physical observables such as total binding energy $E$ (left columns), charge radius $r_{\mathrm{ch}}$ (central columns), and neutron skin $r_{\text{skin}}$ (right columns). Upper row shows the full range of the parameter variations, while the bottom panels magnify the region in the vicinity of optimum parametrization ($f_i=1.0$). On the one end, the range of parameters is limited by the condition that the total energy of the nucleus is negative. On the other end, it is defined by the collapse of numerical solution due to underlying numerical instabilities. For each line, only the indicated parameter is changing while the remaining parameters are kept at the values corresponding to optimum NL5(C) functional.

Figure 2

The masses and coupling constants of the $\sigma$ and $\omega$ mesons in different CEDFs which contain meson exchange. They are combined into three groups dependent on how self- and mixed couplings are introduced. Group A represents the parametrizations which include nonlinear self-couplings only for the $\sigma$ meson. Group B contains the parametrizations which include self-couplings for the $\sigma$ and $\omega$ mesons (and $\rho \mathrm{mesons}$ in the case of PK1R). Group C represents the parametrizations which include density-dependent meson-nucleon couplings for the $\sigma$, $\omega$, and $\rho$ mesons. Note that the mass $m_{\omega }$ of the $\omega$ meson is fixed at indicated values in all functionals except NL3, PK1, and PK1R. The parameters are taken from Refs. [46] (NL1), [49] (NL3), [17] (NL3*), [47] (NL-Z), [58] (NL-Z2), [48] (NL-SH), [59] (NL-RA1), [60] (PK1,PK1R), [61] (TMA), [62] (TM1), [19] (TW99), [63] (DD-ME1), [16] (DD-ME2), [64] (DD-$\mathrm{ME}\delta$), and [60] (PKDD). Note that we omitted mass-dependent terms for $g_{\omega }$ in the TMA parametrization which is a good approximation for heavy nuclei since $g_{\omega }=12.842+3.191{\text{A}}^{-0.4}$ [61]. Vertical solid lines represent the mean values of the respective parameter over the set of indicated functionals. Red dashed and dotted lines show the $\pm 5\%$ and $\pm 10\%$ deviation bands with respect of these mean values.

Figure 3

The same as Fig. 2 but for the $g_{\rho }$ parameter. Red dotted, dashed, and dash-dotted lines show the $\pm 5\%$, $\pm 10\%$, and $\pm 25\%$ deviation bands with respect to mean values.

Figure 4

The same as Fig. 2 but for the $g_2$ and $g_3$ parameters. Red dotted and dash-dotted lines show the $\pm 10\%$ and $\pm 25\%$ deviation bands with respect to the mean values. Note that only the functionals which belong to group A are considered here.

Figure 5

Two-dimensional projections of the distribution of the functional variations in the six-dimensional parameter hyperspace. The colors indicate the $a$ value of the ${\chi }_{\text{norm}}^2\left(\mathbf{p}\right)$ of the functional variation where the latter is expressed as ${\chi }_{\text{norm}}^2\left(\mathbf{p}\right)={\chi }_{\text{norm}}^2\left({\mathbf{p}}_0\right)+a$. The colormap is used for the functional variations with $a\le 1.0$; there are 500 such variations. Note that it was found in the studies that further increase of the number of functional variations does not change the results for statistical errors. In addition, black circles are used for the functionals with $1.0<a\le 1.25$. The optimum functional is shown by large open symbol. The dashed line in panel (a) shows the parametric correlations between $g_2$ and $g_3$ parameters defined by Eq. (10).

Figure 6

The same as Fig. 5 but for the NL5(A) functional. Considering the increase of the size of the parameter hyperspace and extremely time-consuming nature of numerical calculations only 150 functional variations satisfying the condition of Eq. (4) have been collected. This leads to some reduction of the accuracy of the calculations of statistical errors. However, the analysis of the NL5(C) results for which 500 acceptable functional variations have been collected allows us to conclude that statistical errors obtained with the 150 functional variations differ from those calculated with 500 variations only by approximately 5%.

Figure 7

The propagation of statistical errors in binding energies (a), charge radii (b), two-neutron separation energies (c), and neutron skins (d) with neutron number. The results are presented for the Ca ($Z=20$), Ni ($Z=28$), Sn ($Z=50$), and Pb ($Z=82$) isotopes between two-proton and two-neutron drip lines. Open symbols are used to indicate the nuclei whose experimental data of the type shown on the vertical axis of the panel have been used in the fitting protocol of the NL5(C) functional.

Figure 8

The comparison of the differences between calculated and experimental binding energies $E_{\text{th}}-E_{\text{exp}}$ in the Ca, Ni, Sn, and Pb isotopes with relevant statistical errors. The statistical errors obtained with the NL5(C) functional are shown by the shaded area which spans from $-\sigma \left(N\right)$ up to $\sigma \left(N\right)$ at a given value of neutron number $N$. Note that the calculated results for the NL5(C), NL5(D), and NL5(E) functionals are shown only for the nuclei which are spherical in their ground states (see the Appendix) since the statistical errors are defined only for such shapes in the present paper.

Figure 9

The same as in Fig. 8 but for the comparison of the differences between experimental and calculated charge radii $r_{\text{ch}}^{\text{exp}}-r_{\text{ch}}^{\text{th}}$ with relevant statistical errors. The big jump in panel (c) at $N=72$ is due to the deviation of the experimental charge radii of ${}^{122}\mathrm{Sn}$ from the smooth trend seen in the Sn isotopes for $r_{\text{ch}}^{\text{exp}}$ as a function of neutron number [see Fig. 23(b) in Ref. 3].

Figure 10

Statistical errors $\sigma \left(E\right)$ and $\sigma \left(r_{\text{ch}}\right)$ in binding energies (a) and charge radii (b) of the Sn isotopes obtained with the NL5(C) functional compared with respective theoretical systematic uncertainties (spreads) ($\mathrm{\Delta }E$ or $\mathrm{\Delta }{r}_{\text{ch}}$) and systematic errors [“$\sigma {\left(E\right)}_{\text{syst}}$” or “$\sigma {\left(r_{\text{ch}}\right)}_{\text{syst}}$]. The digits 6 and 7 in the labeling of some curves indicate the number of CEDFs used in the definition of theoretical systematic uncertainties and errors. Full set of CEDFs (indicated by 7) includes NL3*, DD-ME2, DD-$\mathrm{ME}\delta$, DD-PC1, NL5(C), NL5(D), and NL5(E) CEDFs. The label 6 is used for the set in which the results of the NL5(C) functional are dropped.

Figure 11

The propagation of statistical errors for the binding energies $\left[\sigma \right(E\left)\right]$, charge radii $\left[\sigma \left(r_{\text{ch}}\right)\right]$, neutron skins $\left[\sigma \left(r_{\text{skin}}\right)\right],$ and two-neutron separation energies $\left[\sigma \left(S_{2n}\right)\right]$ in the Ca and Pb isotope chains. The results of the calculations with the NL5(C) and NL5(A) functionals are shown by solid black line and blue solid line with solid circles, respectively. In addition, red triangle symbols show statistical errors for the NL5(C) functional under the condition that the $g_{\rho }$, $g_2$, and $g_3$ parameters are fixed at the values corresponding to optimum NL5(C) functional.

Figure 12

The differences $E_{\text{th}}-E_{\text{exp}}$ between calculated and experimental binding energies for the indicated CEDFs. The experimental data are taken from Ref. [72] and 830 even-even nuclei, for which measured and estimated masses are available, are included. If $E_{\text{th}}-E_{\text{exp}}<0$, the nucleus is more bound in the calculations than in experiment.

Figure 13

The difference between measured and calculated charge radii $r_{\text{ch}}$ for indicated functionals. The experimental data are taken from Ref. [73].

Figure 14

Charge quadrupole deformations ${\beta }_2$ of the ground states in even-even nuclei obtained in the RHB calculations with indicated CEDFs.