366
8 No-Core Gamow Shell Model
pion mass and Λ the breakdown scale, i.e. the energy after which the model can no
longer be used. It then provides a hierarchy of two-body, three-body, four-body, . . . ,
operators, whose contribution decreases very quickly. In practice, the two-body and
three-body terms have been seen to be the most important. Note that this method also
renormalizes the quark and gluonic effects present in nucleons in the parameters of
the many-body operators of effective field theory.
The first applications of interest have been the N 3 LO (next-to-next-to-next-toleading-order) interaction, which could provide a realistic interaction [4]. Nevertheless, it came at the price of an important three-body term, difficult to take into
account in practice. Hence, it is most of the time approximated by a nucleusdependent two-body Hamiltonian, calculated from the normal ordering of the
Hamiltonian in second quantization. The N 3 LO interaction, however, is problematic
when applied to light nuclei and cannot reproduce the binding energies of the
oxygen isotopes, for example.
Another option was the N 2 LO opt interaction, whose parameters are fitted directly
on nucleon–nucleon phase shifts only. This interaction has the advantage of having
very small three-body forces and can reproduce the binding energies of the oxygen
isotopes. One can also cite the JISP16 (J -matrix inversion of scattering process)
interaction, an effective interaction based on the fit of nucleon–nucleon phase shifts
and nuclear energies up to 16 O, which has no three-body force as well [5].
The three-body force in nuclear effective interactions contains both “initial” and
“induced” components. The “initial” component denotes the three-body force which
appears due to the neglect of quark degrees of freedom and arise from the expansion
of the nucleon–nucleon interaction in the framework of the (chiral) effective field
theory. Indeed, the chiral interaction is written as a power series in p/M, with p
the typical momentum transfer between nucleons and M the mass of a nucleon.
Consequently, high energy degrees of freedom are integrated out in the first terms
of the expansion provided by the effective field theory. This results in the appearance
of many-body forces [6–10]. Added to that, the renormalization such as the V low-k or
the similarity renormalization group (see Refs. [11–14] and discussion afterwards)
also generate the three-nucleon forces. These “induced” three-nucleon forces arise
from the components of the interaction beyond the cutoff Λ and are uniquely related
to the renormalization [11–14].
Typically, nucleon–nucleon interactions are fitted on the properties of nuclei
having A ≤ 4 nucleons. While these very light nuclei are well described therein,
precision deteriorates quickly when considering heavier nuclei [15, 16]. In order to
counter this tendency, the N 2 LO sat (next-to-next-to-leading order with saturation)
interaction has been introduced [17]. The nucleon–nucleon chiral interaction is
considered at N 2 LO level, to which a fit of carbon and oxygen isotopes binding
energies and radii of A = 3, 4 bound nuclei are added. Two and three-body forces
are taken into account in the determination of N 2 LO sat interaction. As a result, the
error made in heavier nuclei is much reduced, as the N 2 LO sat interaction is precise
up to 40 Ca [17].
The study of nucleon–nucleon chiral interactions is still subject to development.
Indeed, one can continue the chiral expansion beyond fourth order, i.e. that of the
8 No-Core Gamow Shell Model
pion mass and Λ the breakdown scale, i.e. the energy after which the model can no
longer be used. It then provides a hierarchy of two-body, three-body, four-body, . . . ,
operators, whose contribution decreases very quickly. In practice, the two-body and
three-body terms have been seen to be the most important. Note that this method also
renormalizes the quark and gluonic effects present in nucleons in the parameters of
the many-body operators of effective field theory.
The first applications of interest have been the N 3 LO (next-to-next-to-next-toleading-order) interaction, which could provide a realistic interaction [4]. Nevertheless, it came at the price of an important three-body term, difficult to take into
account in practice. Hence, it is most of the time approximated by a nucleusdependent two-body Hamiltonian, calculated from the normal ordering of the
Hamiltonian in second quantization. The N 3 LO interaction, however, is problematic
when applied to light nuclei and cannot reproduce the binding energies of the
oxygen isotopes, for example.
Another option was the N 2 LO opt interaction, whose parameters are fitted directly
on nucleon–nucleon phase shifts only. This interaction has the advantage of having
very small three-body forces and can reproduce the binding energies of the oxygen
isotopes. One can also cite the JISP16 (J -matrix inversion of scattering process)
interaction, an effective interaction based on the fit of nucleon–nucleon phase shifts
and nuclear energies up to 16 O, which has no three-body force as well [5].
The three-body force in nuclear effective interactions contains both “initial” and
“induced” components. The “initial” component denotes the three-body force which
appears due to the neglect of quark degrees of freedom and arise from the expansion
of the nucleon–nucleon interaction in the framework of the (chiral) effective field
theory. Indeed, the chiral interaction is written as a power series in p/M, with p
the typical momentum transfer between nucleons and M the mass of a nucleon.
Consequently, high energy degrees of freedom are integrated out in the first terms
of the expansion provided by the effective field theory. This results in the appearance
of many-body forces [6–10]. Added to that, the renormalization such as the V low-k or
the similarity renormalization group (see Refs. [11–14] and discussion afterwards)
also generate the three-nucleon forces. These “induced” three-nucleon forces arise
from the components of the interaction beyond the cutoff Λ and are uniquely related
to the renormalization [11–14].
Typically, nucleon–nucleon interactions are fitted on the properties of nuclei
having A ≤ 4 nucleons. While these very light nuclei are well described therein,
precision deteriorates quickly when considering heavier nuclei [15, 16]. In order to
counter this tendency, the N 2 LO sat (next-to-next-to-leading order with saturation)
interaction has been introduced [17]. The nucleon–nucleon chiral interaction is
considered at N 2 LO level, to which a fit of carbon and oxygen isotopes binding
energies and radii of A = 3, 4 bound nuclei are added. Two and three-body forces
are taken into account in the determination of N 2 LO sat interaction. As a result, the
error made in heavier nuclei is much reduced, as the N 2 LO sat interaction is precise
up to 40 Ca [17].
The study of nucleon–nucleon chiral interactions is still subject to development.
Indeed, one can continue the chiral expansion beyond fourth order, i.e. that of the