152
4 Two-Particle Systems in the Berggren Basis
equation. As a matter of compromise, it has been advocated that the next-to-nextto-next-to-leading order (N 3 LO) chiral interaction is sufficiently precise in practical
applications. This standard approximation will be used in the following to calculate
the ground states of the dineutron, diproton, and deuteron systems.
The theoretical errors made in the derivation of the nuclear forces have been
thoroughly studied over last years. For example, the effects or fifth and sixth orders
in chiral interactions have been considered in Refs. [16, 17], while the effect of
regulators has been assessed in Ref. [18]. Statistical studies of the fit of nucleonnucleon parameters have been done as well using Bayesian inference [19, 20] (see
Sect. 5.10 for more details about the studies of the theoretical limitations of derived
nucleon-nucleon forces).
4.2.1 Numerical Studies of Dineutron, Diproton, and Deuteron in
Berggren Basis
In the following, we will calculate few observable quantities related to dineutron,
diproton, and deuteron, namely the radius of deuteron and the scattering lengths
extracted from nucleon-nucleon scattering cross sections (see Sect. 2.6.6). These
studies are performed in Berggren basis using a variant of the N 3 LO chiral interaction. Actually, the standard N 3 LO chiral interaction [21] cannot be diagonalized
directly in the Berggren basis for two reasons.
Firstly, as it is the renormalized interaction, it depends on an energy cutoff
parameter, beyond which all degrees of freedom are integrated out [22, 23]. The
chiral symmetry breaking scale of the nuclear interaction, occurring for parameters
Λ ∼ 5 fm −1 , corresponding to energies close to 1 GeV, is often used as momentum
cutoff [22–24]. In practice, a momentum cutoff of about 500 MeV is typically used
in the chiral nuclear interaction expansion [24]. However, these energies are too
large for Gamow shell model calculations, as the Berggren basis is suited for low
energies, of the order of a few tens of MeV at most. Consequently, in the following
calculations using Berggren basis, one will use Λ = 1.9 fm −1 , which is a standard
value in many-body nuclear calculations, but where the cutoff dependence is still
present.
Secondly, the N 3 LO interaction is defined in momentum space with propagators
which decrease very slowly in coordinate space. Indeed, used regulators only
suppress high energy components much larger than 1 GeV [24]. As a consequence,
one cannot directly calculate the matrix elements of the Hamiltonian in the Berggren
basis, so that it is necessary to expand the N 3 LO interaction in a basis of a few
harmonic oscillator states, 5–10 typically per partial wave. This additional step
generates basis dependence in Gamow shell model calculations of two-nucleon
systems. Hence , in order to alleviate the latter problems, one has to choose multiply
the N 3 LO interaction by a Fermi function, vanishing after 6–8 fm. Parameters of the
Fermi function, its diffuseness and radius, have been fixed in order to reproduce as
best as possible the basic properties of the dineutron, diproton, and deuteron, such
as the binding energy, the scattering length, and the effective range (see Eqs. (2.185)
4 Two-Particle Systems in the Berggren Basis
equation. As a matter of compromise, it has been advocated that the next-to-nextto-next-to-leading order (N 3 LO) chiral interaction is sufficiently precise in practical
applications. This standard approximation will be used in the following to calculate
the ground states of the dineutron, diproton, and deuteron systems.
The theoretical errors made in the derivation of the nuclear forces have been
thoroughly studied over last years. For example, the effects or fifth and sixth orders
in chiral interactions have been considered in Refs. [16, 17], while the effect of
regulators has been assessed in Ref. [18]. Statistical studies of the fit of nucleonnucleon parameters have been done as well using Bayesian inference [19, 20] (see
Sect. 5.10 for more details about the studies of the theoretical limitations of derived
nucleon-nucleon forces).
4.2.1 Numerical Studies of Dineutron, Diproton, and Deuteron in
Berggren Basis
In the following, we will calculate few observable quantities related to dineutron,
diproton, and deuteron, namely the radius of deuteron and the scattering lengths
extracted from nucleon-nucleon scattering cross sections (see Sect. 2.6.6). These
studies are performed in Berggren basis using a variant of the N 3 LO chiral interaction. Actually, the standard N 3 LO chiral interaction [21] cannot be diagonalized
directly in the Berggren basis for two reasons.
Firstly, as it is the renormalized interaction, it depends on an energy cutoff
parameter, beyond which all degrees of freedom are integrated out [22, 23]. The
chiral symmetry breaking scale of the nuclear interaction, occurring for parameters
Λ ∼ 5 fm −1 , corresponding to energies close to 1 GeV, is often used as momentum
cutoff [22–24]. In practice, a momentum cutoff of about 500 MeV is typically used
in the chiral nuclear interaction expansion [24]. However, these energies are too
large for Gamow shell model calculations, as the Berggren basis is suited for low
energies, of the order of a few tens of MeV at most. Consequently, in the following
calculations using Berggren basis, one will use Λ = 1.9 fm −1 , which is a standard
value in many-body nuclear calculations, but where the cutoff dependence is still
present.
Secondly, the N 3 LO interaction is defined in momentum space with propagators
which decrease very slowly in coordinate space. Indeed, used regulators only
suppress high energy components much larger than 1 GeV [24]. As a consequence,
one cannot directly calculate the matrix elements of the Hamiltonian in the Berggren
basis, so that it is necessary to expand the N 3 LO interaction in a basis of a few
harmonic oscillator states, 5–10 typically per partial wave. This additional step
generates basis dependence in Gamow shell model calculations of two-nucleon
systems. Hence , in order to alleviate the latter problems, one has to choose multiply
the N 3 LO interaction by a Fermi function, vanishing after 6–8 fm. Parameters of the
Fermi function, its diffuseness and radius, have been fixed in order to reproduce as
best as possible the basic properties of the dineutron, diproton, and deuteron, such
as the binding energy, the scattering length, and the effective range (see Eqs. (2.185)