Ab initio charge form factors and radii of light isoscalar nuclei: Role of the two-body charge density

TL;DR

The paper develops ab initio predictions for charge form factors and radii of light isoscalar nuclei ($^4$He, $^6$Li, $^8$Be) using Jacobi-NCSM with chiral SMS interactions and consistently regularized 1N/2N charge densities. It highlights the essential role of two-body charge operators (2N) in shaping FFs at intermediate and large momentum transfers and in achieving accurate charge radii, despite limited impact on the radii from Darwin-Foldy and spin-orbit terms. The authors determine the isospin-1 LEC $M_3$ by fitting to the precise $^4$He radius and propagate LEC uncertainties, finding them negligible for $p$-shell nuclei relative to other sources. They further examine the SRG evolution and two-body relative densities, showing 2N contributions remain sizable and stabilize with system size, indicating that two-body currents are crucial for reconciling ab initio predictions with experimental radii. Overall, the work demonstrates that including consistently regularized two-body charge densities is essential for resolving long-standing underestimations of nuclear charge radii in ab initio approaches and provides a path to extending these predictions to heavier nuclei.

Abstract

We make \textit{ab initio} predictions of charge form factors (FFs) and radii for the isoscalar nuclei $^6$Li and $^8$Be using the Jacobi-coordinate No-Core Shell Model. The calculations employ chiral semilocal momentum-space regularized two- and three-nucleon interactions, together with consistently regularized one- and two-nucleon electromagnetic charge operators. With the short-range charge density fixed to the $^4$He charge radius, the predicted FFs and the $^6$Li radius show good agreement with available experimental data. We find that two-nucleon charge density contributions are essential for describing the FFs, particularly at intermediate and large momentum transfers. Although their influence on the charge radii is limited, these contributions remain crucial for attaining accurate predictions. The present results highlight the importance of two-nucleon charge operators in addressing the long-standing underestimation of nuclear charge radii in \textit{ab initio} calculations based on modern chiral interactions.

Figures (7)

• Figure 1: Charge form factors for $^{4}$He, $^6$Li and $^8$Be with SMS interaction $\Lambda_N=450$ MeV. The experimental data of $^{4}$He are taken from Frosch 1967 Frosch:1967pz, McCarthy 1977 Mccarthy:1977vd, Ottermann 1985 Ottermann:1985km. Data for $^6$Li is from Li 1971 Li:1971tk.
• Figure 2: Charge radii for $^{4}$He, $^6$Li and $^8$Be using the SMS interactions with $\Lambda_N=450$ and $500$ MeV corresponding to the result in Table \ref{['tab:rc']}. The experimental data are indicated by the vertical lines with error bands. $r_{\text{main}}$ (FY) is obtained from the matter radius $r_m$ , while $r_{\text{main}}^{\text{SRG}}$ (J-NCSM) is obtained from the SRG-evolved matter radius $r_m^{\text{SRG}}$; both include the proton and neutron charge radius contributions.
• Figure S1: Charge form factor of $^6$Li from J-NCSM calculations with $\omega=18.0$ MeV and different $N_\mathrm{HO}$. The adopted interaction is SMS N$^{4}$LO$^{+}$ + N$^{2}$LO $\Lambda_N=$450 MeV evolved to flow parameters 1.88, 2.236 and 2.600 fm$^{-1}$. The dotted lines present the results with one-body operators and solid lines for cases considering the 2N operators. Experimental data are taken from Ref. Li:1971tk.
• Figure S2: Charge form factor of $^6$Li from J-NCSM calculations with $\omega=20.0$ MeV and different $N_\mathrm{HO}$. The adopted interaction is SMS N$^{4}$LO$^{+}$ + N$^{2}$LO $\Lambda_N=$450 MeV evolved to flow parameter 1.88, 2.236, and 2.600 fm$^{-1}$. The dotted lines represent the results with one-body operators and solid lines for cases considering the 2N operators. Experimental data are taken from Ref. Li:1971tk.
• Figure S3: Contribution from two-nucleon contact and one-pion-exchange terms to the charge radii of $^6$Li from J-NCSM calculations with different HO parameters. The adopted interaction is SMS N$^{4}$LO$^{+}$ + N$^{2}$LO $\Lambda_N=$450 MeV evolved to flow parameter 1.88, 2.236 and 2.600 fm$^{-1}$.