4.2 Study of Two-Nucleon Systems with Realistic Interactions
153
and (2.186)). By taking a diffuseness of 0.65 fm, and a radius of 7.5 fm for dineutron
and deuteron, and 6.85 fm for diproton, it has been possible to reproduce these
observables with an error of 1–5% in comparison to N 3 LO calculations, where
the Schrödinger equation is solved up to machine precision, in which Λ = 4 − 6
fm −1 [24]. The Fermi radius for diproton has a different value compared to that of
dineutron and deuteron due to the presence of the Coulomb interaction. Note that
one takes the bare Coulomb interaction, in order to be consistent with the use of
Coulomb wave functions to generate the diproton Berggren basis.
At most two channels have to be included to study eigenstates of the two-nucleon
systems. These channels consist of the spin singlet (S = 0) and spin triplet (S = 1)
channels. Due to the Pauli principle, 0 + states of two-nucleon systems have a T = 1
isospin and reduce to a single channel, which is the 1 S 0 channel. Conversely, the
orbital angular momenta = 0, 2 are present in the partial decomposition of the 1 +
state of two-nucleon systems, consisting in the 3 S 1 and 3 D 1 channels. The 1 + state
has isospin T = 0 due to the Pauli principle.
Due to the strong binding character of the nucleon-nucleon T = 0 interaction,
the nucleon-nucleon interaction is sufficiently strong to bind proton and neutron.
The 1 + state in a deuteron is, in fact, the only bound state of two-nucleon systems.
Conversely, the T = 1 interaction is weaker than the T = 0 interaction, so that
the 0 + states of two-nucleon systems are always unbound. Due to the absence of
Coulomb and centrifugal barrier for dineutron and deuteron in the 1 S 0 channel, 0 +
is an antibound state. However, even though the the T = 1 interaction cannot bind
two nucleons, its attractive character is not negligible, as that the 0 + antibound state
of dineutron and deuteron has a small binding energy. In the diproton case, the small
Coulomb barrier implies, as stated before, that it is an S-matrix pole of negative
energy but with a positive width.
One cannot generate experimentally an antibound state or an unbound state of
negative energy and positive width, as they do not correspond to the physical states,
which are either bound states, narrow resonances, or scattering states. Consequently,
to infer the presence or absence of bound or unbound S-matrix poles close to the
particle-emission threshold, one proceeds indirectly by considering the scattering
length of the two-body system at zero energy (see Eqs. (2.185) and (2.186)). The
measured scattering length is negative for the 0 + states and positive for the 1 + state,
so that the unbound character of 0 + states and bound character of the 1 + state can
be inferred from it.
One will now present results obtained with the Berggren basis expansion for
the S-matrix poles of dineutron, diproton, and deuteron. The Berggren basis is
generated by Woods-Saxon potentials with the diffusivity d = 0.65 fm, the
radius R 0 = 1.5 fm, and the depth of central part V 0 = 30 MeV (0 + states) or
V 0 = 40 MeV (1 + states). The 1 S 0 channel for dineutron and diproton contains
an antibound pole of the S-matrix with negative energy and positive width, while
the 3 S 1 and 3 D 1 channels contain a loosely bound state, but no narrow resonance.
The Berggren contours have to be complex to expand antibound states and S-matrix
poles of negative energy and positive width in the 1 S 0 channel. They consist of
three segments defined by k = 0, k = −0.1 − 0.1i fm −1 , k = 0.2 − 0.1i fm −1 , and
153
and (2.186)). By taking a diffuseness of 0.65 fm, and a radius of 7.5 fm for dineutron
and deuteron, and 6.85 fm for diproton, it has been possible to reproduce these
observables with an error of 1–5% in comparison to N 3 LO calculations, where
the Schrödinger equation is solved up to machine precision, in which Λ = 4 − 6
fm −1 [24]. The Fermi radius for diproton has a different value compared to that of
dineutron and deuteron due to the presence of the Coulomb interaction. Note that
one takes the bare Coulomb interaction, in order to be consistent with the use of
Coulomb wave functions to generate the diproton Berggren basis.
At most two channels have to be included to study eigenstates of the two-nucleon
systems. These channels consist of the spin singlet (S = 0) and spin triplet (S = 1)
channels. Due to the Pauli principle, 0 + states of two-nucleon systems have a T = 1
isospin and reduce to a single channel, which is the 1 S 0 channel. Conversely, the
orbital angular momenta = 0, 2 are present in the partial decomposition of the 1 +
state of two-nucleon systems, consisting in the 3 S 1 and 3 D 1 channels. The 1 + state
has isospin T = 0 due to the Pauli principle.
Due to the strong binding character of the nucleon-nucleon T = 0 interaction,
the nucleon-nucleon interaction is sufficiently strong to bind proton and neutron.
The 1 + state in a deuteron is, in fact, the only bound state of two-nucleon systems.
Conversely, the T = 1 interaction is weaker than the T = 0 interaction, so that
the 0 + states of two-nucleon systems are always unbound. Due to the absence of
Coulomb and centrifugal barrier for dineutron and deuteron in the 1 S 0 channel, 0 +
is an antibound state. However, even though the the T = 1 interaction cannot bind
two nucleons, its attractive character is not negligible, as that the 0 + antibound state
of dineutron and deuteron has a small binding energy. In the diproton case, the small
Coulomb barrier implies, as stated before, that it is an S-matrix pole of negative
energy but with a positive width.
One cannot generate experimentally an antibound state or an unbound state of
negative energy and positive width, as they do not correspond to the physical states,
which are either bound states, narrow resonances, or scattering states. Consequently,
to infer the presence or absence of bound or unbound S-matrix poles close to the
particle-emission threshold, one proceeds indirectly by considering the scattering
length of the two-body system at zero energy (see Eqs. (2.185) and (2.186)). The
measured scattering length is negative for the 0 + states and positive for the 1 + state,
so that the unbound character of 0 + states and bound character of the 1 + state can
be inferred from it.
One will now present results obtained with the Berggren basis expansion for
the S-matrix poles of dineutron, diproton, and deuteron. The Berggren basis is
generated by Woods-Saxon potentials with the diffusivity d = 0.65 fm, the
radius R 0 = 1.5 fm, and the depth of central part V 0 = 30 MeV (0 + states) or
V 0 = 40 MeV (1 + states). The 1 S 0 channel for dineutron and diproton contains
an antibound pole of the S-matrix with negative energy and positive width, while
the 3 S 1 and 3 D 1 channels contain a loosely bound state, but no narrow resonance.
The Berggren contours have to be complex to expand antibound states and S-matrix
poles of negative energy and positive width in the 1 S 0 channel. They consist of
three segments defined by k = 0, k = −0.1 − 0.1i fm −1 , k = 0.2 − 0.1i fm −1 , and