8.4 Renormalization of Realistic Interactions in a Finite Model Space Using. . .
393
8.4.1 Application of the ˆ ¯
Q-Box Method in Oxygen and Fluorine
Chains of Isotopes
The first application of ˆ ¯
Q-box method for Gamow shell model calculations can
be found in Ref. [139]. Below, we will discuss briefly most important features of
these calculation for the isotopes of oxygen and fluorine. The considered nucleon–
nucleon interaction is the optimized chiral interaction NNLO opt , which offers a good
description of nuclear structure, i.e. binding energies, excitation spectra, positions of
driplines and neutron matter equation of state, without the need of three-body forces
[129]. Figure 8.9 compares the Gamow shell model ground state energies (E gs ) and
single-neutron separation energies (S n ) in the oxygen and fluorine isotopic chains
with experimental data and other theoretical calculations. The neutron dripline of
oxygen isotopes, situated at 24 O, has been reproduced in shell model calculations
[144], thanks to the repulsive character of three-nucleon forces (see curves for
shell model with two- and three-nucleon forces in Fig. 8.9). All calculations which
include the effects of three-nucleon forces predict the correct position of the neutron
dripline, as shown in Fig. 8.9. However, quantitative results vary widely, which is
the case, in particular, for nuclei beyond the neutron dripline. This might arise from
the fact that open decay channels and particle continuum are not always included,
as it occurs in standard shell model with USDB interaction [145], shell model
with two-nucleon and three-nucleon forces [144], or valence space in-medium
similarity renormalization group, also with two-nucleon forces and three-nucleon
forces [143]. Indeed, in the shell model embedded in the continuum studies of
oxygen and fluorine chains of isotopes, it was shown that the position of driplines
can be described with standard two-body interaction when the coupling to the decay
channels is included [147, 148]. Hence, the effects of three-nucleon forces and the
coupling to the scattering continuum can hardly be disentangled. The Gamow shell
model approach, hereby based on many-body perturbation theory using two-body
interactions, provides with the correct location for the neutron dripline of oxygen
isotopes. One also obtains a good description of the 25,26 O isotopes, which are
situated beyond the neutron dripline (see the left panel of Fig. 8.9).
Let us consider the two other calculations employing the Berggren representation
presented in Fig. 8.9. The first calculation corresponds to the coupled-cluster model
using Berggren basis [141] and a schematic three-nucleon force. In this case the
neutron dripline is reproduced, but results deviate from experimental data. The
second calculation is that of the Gamow shell model [142], where an effective twobody interaction fitted to the bound states and resonances of 23–26 O is utilized. One
assumes that the core of 22 O is inert. A proper description of energies is obtained
therein. It is also predicted that the 28 O ground state is very loosely bound or weakly
unbound. Moreover, the 28 O isotope is still more bound than 27 O, while being
slightly unbound with respect to 26 O.
The unbound character of the 25–28 O isotopes is clearly seen in the one neutron
separation energies (see Fig. 8.9). The issue of tetraneutron radioactivity in 28 O
was raised in Ref. [149]. A direct tetraneutron decay could occur if S 4n < 0 and
393
8.4.1 Application of the ˆ ¯
Q-Box Method in Oxygen and Fluorine
Chains of Isotopes
The first application of ˆ ¯
Q-box method for Gamow shell model calculations can
be found in Ref. [139]. Below, we will discuss briefly most important features of
these calculation for the isotopes of oxygen and fluorine. The considered nucleon–
nucleon interaction is the optimized chiral interaction NNLO opt , which offers a good
description of nuclear structure, i.e. binding energies, excitation spectra, positions of
driplines and neutron matter equation of state, without the need of three-body forces
[129]. Figure 8.9 compares the Gamow shell model ground state energies (E gs ) and
single-neutron separation energies (S n ) in the oxygen and fluorine isotopic chains
with experimental data and other theoretical calculations. The neutron dripline of
oxygen isotopes, situated at 24 O, has been reproduced in shell model calculations
[144], thanks to the repulsive character of three-nucleon forces (see curves for
shell model with two- and three-nucleon forces in Fig. 8.9). All calculations which
include the effects of three-nucleon forces predict the correct position of the neutron
dripline, as shown in Fig. 8.9. However, quantitative results vary widely, which is
the case, in particular, for nuclei beyond the neutron dripline. This might arise from
the fact that open decay channels and particle continuum are not always included,
as it occurs in standard shell model with USDB interaction [145], shell model
with two-nucleon and three-nucleon forces [144], or valence space in-medium
similarity renormalization group, also with two-nucleon forces and three-nucleon
forces [143]. Indeed, in the shell model embedded in the continuum studies of
oxygen and fluorine chains of isotopes, it was shown that the position of driplines
can be described with standard two-body interaction when the coupling to the decay
channels is included [147, 148]. Hence, the effects of three-nucleon forces and the
coupling to the scattering continuum can hardly be disentangled. The Gamow shell
model approach, hereby based on many-body perturbation theory using two-body
interactions, provides with the correct location for the neutron dripline of oxygen
isotopes. One also obtains a good description of the 25,26 O isotopes, which are
situated beyond the neutron dripline (see the left panel of Fig. 8.9).
Let us consider the two other calculations employing the Berggren representation
presented in Fig. 8.9. The first calculation corresponds to the coupled-cluster model
using Berggren basis [141] and a schematic three-nucleon force. In this case the
neutron dripline is reproduced, but results deviate from experimental data. The
second calculation is that of the Gamow shell model [142], where an effective twobody interaction fitted to the bound states and resonances of 23–26 O is utilized. One
assumes that the core of 22 O is inert. A proper description of energies is obtained
therein. It is also predicted that the 28 O ground state is very loosely bound or weakly
unbound. Moreover, the 28 O isotope is still more bound than 27 O, while being
slightly unbound with respect to 26 O.
The unbound character of the 25–28 O isotopes is clearly seen in the one neutron
separation energies (see Fig. 8.9). The issue of tetraneutron radioactivity in 28 O
was raised in Ref. [149]. A direct tetraneutron decay could occur if S 4n < 0 and