Impact of tensor couplings with scalar mixing on covariant energy density functionals

The recent pioneering campaigns conducted by the Lead Radius Experiment (PREX) and the Calcium Radius Experiment (CREX) Collaborations have uncovered major deficiencies in the theoretical description of some fundamental properties of atomic nuclei. Following a recent refinement to the isovector sector of covariant energy density functionals by Reed et al. [Phys. Rev. C 109, 035803 (2024)], we present here additional improvements to the functional by including both tensor couplings and an isoscalar-isovector mixing term in the scalar sector. Motivated by the distinct surface properties of calcium and lead, we expect that the tensor terms that generate derivative couplings will help break the linear correlation between the neutron skin thickness of these two nuclei. Moreover, the addition of these new terms mitigates most of the problems identified by Reed et al. in describing the properties of both finite nuclei and neutron stars. While significant progress has been made in reconciling the PREX-CREX results without compromising other observables, the final resolution awaits the completion of a proper calibration for this new class of functionals. We expect that powerful reduced basis methods used recently to create efficient emulators will be essential to accomplish this task.

Figure 1

Point proton (dashed lines) and neutron (solid lines) densities of ${}^{208}\mathrm{Pb}$ as predicted by the three DINO models [21], alongside predictions from the FSU-TD0 model introduced in this work (see Table 4). The FSU-TD0 model eliminates the large oscillations in the nuclear interior by trading the $\delta$-meson term in the DINO models in favor of a $\rho$-meson tensor interaction.

Figure 2

Charge density of ${}^{48}\mathrm{Ca}$ (red) and ${}^{208}\mathrm{Pb}$ (green) as predicted by the DINOa model [21] with and without the inclusion of a tensor-coupled $\rho$ meson. For $f_{\rho }=-4$ (dot-dashed line), the density fluctuations in the nuclear interior increase. In contrast, a positive $f_{\rho }=10$ (dashed line) mitigates the charge density fluctuations in the interior, thereby improving the agreement with experiment (circled points) [57]. Predictions from the original DINOa model are depicted with a solid line.

Figure 3

Charge density distributions for ${}^{48}\mathrm{Ca}$ (red) and ${}^{208}\mathrm{Pb}$ (green). The effect of adding the two tensor interactions is seen to reduce the fluctuations in the nuclear interior that are characteristic of the DINO models.

Figure 4

Impact of the scalar mixing term on both neutron star radii and the weak-skin form factor of ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ (see inset in the upper right panel). The predictions of the three DINO models for the weak-skin form factors are shown along with the results obtained after adding a positive (green) and negative (red) scalar mixing term of magnitude ${\mathrm{\Lambda }}_{\mathrm{s}}=\pm 0.00025$. The effects on stellar radii and the tidal deformability are displayed in the figure and the labels, respectively.

Figure 5

Charge density distributions for ${}^{48}\mathrm{Ca}$ (left) and ${}^{208}\mathrm{Pb}$ (right) as predicted by the three new FSU-TD models, alongside the predictions from FSUGarnet [37]. Shown in the inset is the square of the charge form factor $|F_{\mathrm{ch}}{|}^2\left(q\right)$ as a function of the momentum transfer. The experimental charge form factors and corresponding spatial densities are obtained from Ref. [57].

Figure 6

Weak skin form factor $F_{\mathrm{ch}}-F_{\mathrm{wk}}$ for the new class of FSU-TD models displayed with stars in the left panel alongside other RMF models, such as FSUGarnet and FSUGold2. Shown together with the three DINO models labeled a,b,c are predictions from FSU-TD0 (green), FSU-TD1 (blue), and FSU-TD2 (red). The right panel shows the mass-radius relations as predicted by the various models and include in parenthesis, the tidal deformabilities for a $M=1.4M_{\odot }$ neutron star. Also included are the $68\%$ and $95\%$ confidence intervals for the two NICER sources J0030 and J0740.

Figure 7

Weak skin form factor $F_{\mathrm{ch}}-F_{\mathrm{wk}}$ as a function of momentum transfer $q$ for the new class of FSU-TD models. Also shown with their associated colors are predictions from the FSUGold2 [36] and FSUGarnet [37] models, alongside the CREX and PREX results. Note that while FSUGold2 pins down the PREX result, it dramatically overestimates the CREX result. In contrast, the softer FSUGarnet model shows some improvement for CREX but underestimates the PREX result.

Figure 8

The energy per neutron as a function of density. Results from $\chi \mathrm{EFT}$ [61] are shown in light blue with their associated uncertainty bands. The predictions of the three FSU-TD models are given with (dashed lines) and without (solid lines) the $\xi$ term. The $\xi$ coupling constant has no noticeable impact on the predictions of the FSU-TD0 model, so it is therefore omitted.

Figure 9

Mass-radius relations for the FSU-TD models (solid lines) and the FSU-TD models supplemented by the addition of the quartic, isovector term $\xi$ (dashed lines). Also shown are NICER's $68\%$ and $95\%$ confidence contours together with predictions for the tidal deformabilities of a $M=1.4M_{\odot }$ neutron star (see legends).

Figure 10

The symmetry energy calculated for some of the new classes of models with enlarged isovector sectors. The result using the parabolic approximation (dashed) differs—in some instances significantly—from the result using the exact expression for the symmetry energy (solid) depending on the strength of the isovector coupling constants.