Ab initio computations of the fourth-order charge density moments of $^{48}$Ca and $^{208}$Pb

TL;DR

We address how high-precision measurements of charge-density moments can inform neutron radii in neutron-rich nuclei. Using ab initio IMSRG calculations with chiral EFT Hamiltonians, the study computes charge form factors for ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ and establishes strong correlations between $R_\mathrm{ch}^4$, $R_\mathrm{ch}^2$, and $R_\mathrm{n}^2$, enabling predictions for $R_\mathrm{ch}^4$ that agree with previous extractions. It finds a comparatively weak connection between $R_\mathrm{ch}^4$ and the neutron skin $R_\mathrm{skin}$, limiting model-independent extraction of skins from $R_\mathrm{ch}^4$ alone, though combined radii constraints and complementary measurements remain informative. The work emphasizes the role of precise $F_\mathrm{ch}(q^2)$ data and Gaussian-process methods for robust moment extraction and provides reference values for upcoming electron-scattering experiments.

Abstract

Neutron skins of neutron-rich nuclei connect nuclei with the matter in neutron stars. High-precision measurements of nuclear charge densities to extract higher-order moments are proposed to be sensitive to neutron radii and skin thicknesses. We investigate the charge density of $^{48}$Ca and $^{208}$Pb, leading candidates for such studies, with ab initio nuclear structure calculations. We find strong correlations between the fourth-order charge density moment $R_\mathrm{ch}^4$ and the charge and neutron radii, allowing us to predict $R_\mathrm{ch}^4$ for $^{48}$Ca and $^{208}$Pb. We find a substantially weaker correlation between the fourth-order charge density moment and the neutron skin, limiting the ability of high-precision electron scattering to determine the neutron skin in a model-independent manner.

Figures (3)

• Figure 1: Predictions of $R_\mathrm{ch}^2$, $R_\mathrm{ch}^4$, and $R_\mathrm{n}^2$ (for ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ on the left and right, respectively) using the $\Delta\mathrm{NNLO}_\mathrm{GO}$Jiang2020PRC_DN2LOGO and $\mathrm{1.8/2.0}~\mathrm{(EM7.5)}$Arthuis2024arxiv_LowResForces Hamiltonians and the 34 nonimplausible Hamiltonians from Ref. Hu2022NP_Pb208. We show best fit lines to the correlations between $R_\mathrm{ch}^4$, $R_\mathrm{ch}^2$, and $R_\mathrm{n}^2$ for the nonimplausible Hamiltonians, each with a conservative error band. The Pearson correlation coefficient $r$ for each correlation is given in the top left. Using these correlations and experimental determinations of $R_\mathrm{ch}^2$ for ${}^{48}\mathrm{Ca}$Noel2024JHEP_ChargeDensities and ${}^{208}\mathrm{Pb}$Kurasawa2021PTEP_R4Measurement and a theoretical prediction of $R_\mathrm{n}^2$Heinz2024inprep_MuToEResponses, we provide predictions for $R_\mathrm{ch}^4$ of ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$.
• Figure 2: Predictions of $R_\mathrm{skin}$ and $R_\mathrm{ch}^4$ (for ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ on the left and right, respectively) using the $\Delta\mathrm{NNLO}_\mathrm{GO}$Jiang2020PRC_DN2LOGO and $\mathrm{1.8/2.0}~\mathrm{(EM7.5)}$Arthuis2024arxiv_LowResForces Hamiltonians and the 34 nonimplausible Hamiltonians from Ref. Hu2022NP_Pb208. We compare with our determinations of $R_\mathrm{ch}^4$ from Fig. \ref{['fig:rch4_correlation']}, which are based on experimental $R_\mathrm{ch}^2$ measurements Kurasawa2021PTEP_R4MeasurementNoel2024JHEP_ChargeDensities and a theoretical $R_\mathrm{n}^2$ prediction for ${}^{48}\mathrm{Ca}$Heinz2024inprep_MuToEResponses. We also compare with literature values for $R_\mathrm{skin}$ (above), from parity-violating electron scattering (CREX/PREX, black) Adhikari2021Adhikari2022, electric dipole responses (Birkhan et al./Tamii et al., pink) Tamii2011PRLBirkhan2017PRL, hadronic probes (Kłos et al, Zenihiro et al., red) Klos2007Zenihiro2010, ab initio predictions (Hagen et al., brown; Hu et al., orange; Heinz et al., purple) Hagen2016NP_Ca48SkinHu2022NP_Pb208Heinz2024inprep_MuToEResponses, nonrelativistic density-functional theory (DFT) predictions using the SV-min functional (Reinhard et al., olive) Reinhard2021PRLReinhard2022PRL, and inferences from experimental data based on relativistic and nonrelativistic DFT correlations (Kurasawa et al., light and dark gray, respectively) Kurasawa2021PTEP_R4Measurement.
• Figure 3: Point-proton form factor (left), point-proton mean-squared radius $R_\mathrm{p}^2$ (middle), and fourth-order moment of point-proton density distribution $R_\mathrm{p}^4$ (right) for $^{3}$He. In the left panel, the form factor and its first and second derivatives are shown. The circles are points used to train the GP, whose 95% confidence intervals are given by shaded areas. The dashed curves are computed with cubic spline interpolation. In the middle panel, $R_\mathrm{p}^2$ are computed with the GPs trained with the $n$ lowest $q^{2}$ points, and the $68\%$ and $95\%$ confidence intervals are given by the bars. The solid, dashed, and dotted lines are computed with the $R_\mathrm{p}^2$ operator, slope of the spline interpolation, and the radial integral of the density obtained with the spline interpolation, respectively. The right panel is the same except for $R_\mathrm{p}^4$.