3.1.3. GSM Calculations with Phenomenological Nuclear Potential

Many ab initio calculations, such as SM [82], VS-IMSRG [81], complex CC [31], and realistic GSM [38,39,41] calculations, have been employed for the description of neutronrich oxygen and fluorine isotopes. However, these calculations bear a large theoretical uncertainty. Furthermore, results arising from ab initio calculations depend on the realistic nuclear forces used (a short summary of the VS-IMSRG calculations based on different chiral nuclear forces can be found in Ref. [80]). Moreover, continuum coupling is absent in the VS-IMSRG [81] and SM [82] calculations. Similar situations also occur for neutron-rich fluorine isotopes, where few calculations have been performed and most of the calculations are absent for the continuum coupling [83,84]. Based on these grounds, we performed the GSM calculations with a phenomenological nuclear interaction for neutron-rich oxygen and fluorine drip line nuclei.

**Figure 5.** Energies of ground states in <sup>24</sup>−28O, calculated by GSM within the *sdpf* (*s*1/2, *p*1/2,3/2, *d*3/2,5/2, *f*5/2,7/2 partial waves) (**upper**) and *sd* (*s*1/2, *d*3/2,5/2 partial waves) (**lower**) model spaces, using the effective field theory (EFT) EFT(318)(the value within braket stands momentum cutoffs), EFT(356), EFT(390), EFT(436), and Furutani–Horiuchi–Tamagaki (FHT) interactions, with A-independence (*A*-indep) or A-dependence (*A*-dep) (see details in Ref. [42]). Results are compared with the experimental data available, represented by a star. The data for 25,26O and 27,28O are taken from experiment (see Refs. [8,74]) and evaluations given in AME2016 [78] (with permissions from Ref. [42]).

For the considered neutron-rich oxygen isotopes, the closed-shell nucleus 22O is selected to be the inner core. The one-body potential is mimicked by a WS potential, whose parameters are adjusted to reproduce the single-particle spectrum of 23O [10]. We use the pionless EFT interaction [85,86] as the two-body interaction. Owing to the few available data related to the oxygen drip line nuclei [10], only the leading order (LO) NN interaction of the EFT force is fitted to reproduce selected experimental data. The effect of the 3NF at LO is then effectively taken into account in the fitted parameters. Details can be found in Ref. [42]. We calculated the energies of the ground states of <sup>24</sup>−28O with GSM within *sdpf* (*s*1/2, *p*1/2,3/2, *d*3/2,5/2, *f*5/2,7/2 partial waves) and *sd* (*s*1/2, *d*3/2,5/2 partial waves) active spaces, using different EFT interactions (see details in Ref. [42]). The calculated groundstate energies of <sup>24</sup>−28O are shown in Figure 5. The calculations within the *sd* space show that the <sup>25</sup>−28O isotopes are unbound and that their binding energies are close to the experimental data [8,74,78] and to calculations performed within the *sdpf* space. However, the calculations obtained in the *sd* space provide an unbound 26O ground state, by about 300 keV relative to the ground state of 24O, which is a little higher than its experimental value, which is about 20 keV unbound [8]. Though the energy difference obtained using the two different model spaces is small, the calculation performed within the *spdf* space seems to be more reasonable. The GSM calculations performed within the *sdpf* space provide good agreements of the <sup>23</sup>−26O ground states with experimental data [8,74,78]; in particular, the two-neutron separation energy (*S*2*n*) of 26O is about 20 keV [8]. The calculated ground state of 28O is unbound in all three cases and located about 700 keV above the ground state of 24O. The ground states of the 26,28O isotopes are unbound, but bear negligible widths. Together with the calculated one-body densities of the 26,28O isotopes in Ref. [42], the results suggest that the ground state of 28O exhibits four-neutron decay by way of 2*n*-2*n* emission via the 26O ground state, which is consistent with few-body [87] and the above ab initio GSM calculations [41].

**Figure 6.** Energies of <sup>25</sup>−31F with respect to the 24O core, calculated within different theoretical frameworks and compared to experimental data [78]. Besides the GSM calculations using FHT and EFT interactions, calculations in the Hartree-Fock many-body perturbation theory (HF-MBPT) [88] and VS-IMSRG [84] frameworks, utilizing the harmonic-oscillator (HO) basis, hence being without continuum coupling, are presented (with permissions from Ref. [18]).

Figure 6 shows the GSM calculations of the binding energies of fluorine isotopes using FHT and EFT interactions (see details in Ref. [18]), wich are compared with experimental data [78] and Hartree-Fock MBPT (HF-MBPT) [88] and VS-IMSRG [84] calculations, which are both performed in the HO basis. The energy of 25F has been fixed to its experimental datum in all used models in Figure 6. We can see that all calculations reproduce the ground state energies of <sup>25</sup>−28F isotopes well, situated in the well-bound region, whereas differences start after 29F, i.e., when one reaches the neutron drip line. Due to the absence of both multi-shell and continuum couplings, the VS-IMSRG calculations [84] provide visible differences, which are about 4- to 5-MeV in magnitude for 30,31F. When applying the HF-MBPT method [88], the cross-shell couplings generated by the *sd* and *p f* shells are included, so that proper binding energies of up to 29F are predicted. However, due to the lack of continuum coupling, the binding energies of 30,31F are about 1 MeV away from experimental data. The GSM calculation performed with FHT and EFT interactions correctly provides binding energies of up to 31F. Moreover, the odd–even staggering encountered from the 28F isotope, typical of the presence of a strong proton–neutron interaction, is well reproduced, with 30F being unbound and 31F being loosely bound in our calculations.

Recent realistic shell model calculations [89] have pointed out that nuclear deformation plays an important role in the neutron drip line nuclei. Within GSM, deformation can be accounted for by configuration mixing using a cross-shell valence space [5]. Besides deformation, continuum coupling also gives important contributions in drip line nuclei. They are strongly coupled with continuum states near the particle-emission threshold, which provides additional binding energy [5]. This situation is unlike that occurring in well-bound systems, where one only has strong coupling with nearby deeply-bound single-particle states.

**Figure 7.** One-nucleon densities of the bound 27,29,31F isotopes calculated with the GSM using the EFT interaction in the valence space as a function of radii *r*, respectively, depicted by short-dashed, long-dashed, and solid lines. The rms radii of these isotopes are shown in the inset (with permissions from Ref. [18]).

The two-neutron separation energy *S*2*<sup>n</sup>* of 31F is about 170 keV [78], which is sufficiently small for sustaining a two-neutron halo. We calculated the one-nucleon densities and neutron rms radii of the neutron-bound 27,29,31F isotopes using GSM with the EFT interaction (see Figure 7). From our calculations, a halo clearly develops in the asymptotic region of 31F. Indeed, on the one hand, the one-nucleon density of 31F slowly decreases on the real axis and is about one to two orders of magnitude larger than those of 27,29F in

the asymptotic region. Added to that, on the other hand, the neutron rms radius of 31F does not follow the trend noticed in 27,29F, as the rms radius sharply increases compared to the associated values in 27F and 29F. Consequently, one can assume from these GSM calculations [18] that 31F is a two-neutron halo state.
