Ground-state properties of doubly magic nuclei from the unitary-model-operator approach with chiral two- and three-nucleon forces

The ground-state energies and radii for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ are calculated with the unitary-model-operator approach (UMOA). In the present study, we employ similarity renormalization group (SRG) evolved nucleon-nucleon ($NN$) and three-nucleon ($3N$) interactions based on chiral effective field theory. This is the first UMOA calculation with both $NN$ and $3N$ interactions. The calculated ground-state energies and radii are consistent with the recent ab initio results with the same interaction. We evaluate the expectation values of two- and three-body SRG evolved radius operators, in addition to those of the bare radius operator. With the aid of the higher-body evolution of the radius operator, it is seen that the calculated radii tend to be SRG resolution-scale independent. We find that the SRG evolution gives minor modifications for the radius operator.

Figure 1

Hugenholtz diagrams for the ground-state energy up to the third order. Note that the first order contributions are omitted. The cross and dot indicate the one- and two-body part of Hamiltonian, respectively. The diagram rules are same as in Ref. [41].

Figure 2

Ground-state energies for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ as functions of $\hslash \omega$ with the $NN+3N$–full interaction. The dashed (solid) lines calculated without (with) the three-body cluster term energy. The interaction is obtained by SRG evolution of chiral ${\mathrm{N}}^3\mathrm{LO}NN$ [51] and ${\mathrm{N}}^2\mathrm{LO}3N$ [6, 39] interactions up to ${\lambda }_{\mathrm{SRG}}=1.88{\mathrm{fm}}^{-1}$.

Figure 3

Ground-state energies from the Hartree-Fock basis many-body perturbation theory (HF-MBPT), UMOA, and coupled-cluster method (CCM). The energies of HF-MBPT are calculated with Eqs. (31, 32, 33). The CCM results are taken from Refs. [18, 56]. The interaction is obtained by SRG evolution of chiral ${\mathrm{N}}^3\mathrm{LO}NN$ [51] and ${\mathrm{N}}^2\mathrm{LO}3N$ [6, 39] interactions up to ${\lambda }_{\mathrm{SRG}}=1.88{\mathrm{fm}}^{-1}$. For the HF-MBPT and UMOA energies, $E^{1,2\mathrm{BC}}$ (square), $E^{3\mathrm{BC}}$ (star), and $E_{\mathrm{g}.\mathrm{s}.}$ (circle) are shown. For the CCM energies, CCSD (down triangle), triple correction (up triangle), and CR-CC(2,3) energies (pentagon) are shown.

Figure 4

Ground-state energies for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$. All the calculation results are obtained at $e_{max}=14$ and $\hslash \omega =25$ MeV. The experimental data are taken from Ref. [50].

Figure 5

Expectation values of bare root-mean-squared radius operator for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ as functions of $\hslash \omega$. Here, $NN+3N$–full interaction at ${\lambda }_{\mathrm{SRG}}=1.88{\mathrm{fm}}^{-1}$ is employed.

Figure 6

Charge radii for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$. All calculation results are obtained at $\hslash \omega =25$ MeV and $e_{max}=14$. To evaluate the charge radii, the bare mean-squared radius operators are used. The experimental data are taken from Ref. [58].

Figure 7

Charge radii evaluated with the expectation values of bare, two-body (2B) SRG evolved, and three-body (3B) SRG evolved mean-squared radius operator for ${}^4\mathrm{He}$ [(a) and (d)], ${}^{16}\mathrm{O}$ [(b) and (e)], and ${}^{40}\mathrm{Ca}$ [(c) and (f)]. All calculation results are obtained at $\hslash \omega =25$ MeV and $e_{max}=14$. The calculation results in (a), (b), and (c) are calculated with $NN+3N$–ind interactions and the calculation results in (d), (e), and (f) are calculated with $NN+3N$–full interactions.

Figure 8

Hugenholtz diagrams for the ground-state energy at the fourth order. The diagrams are classified into single, double, triple, and quadruple topologies, depending on the number of intermediate particle-hole excitations. The diagram rules are same as in Ref. [41].