4.2 Study of Two-Nucleon Systems with Realistic Interactions
155
The energies and widths obtained for the unbound 0 + state of dineutron,
diproton, and deuteron are of particular interest, as they usually cannot be calculated
with standard methods. One can see that one obtains these unbound states at energy
close to zero, in accordance with experimental data. The real part of the energy of the
diproton is nearly the same as that of the dineutron or the excited T = 1 deuteron
state, whereas its imaginary part is close to 1 MeV in absolute value. This shows
that the Coulomb interaction present in the Hamiltonian can drastically change the
character of S-matrix pole of the two-body systems. In fact, the diproton ground
state is not a resonance, but a virtual state, as it has a negative energy and a positive
width.
An antibound or virtual state cannot be associated to a physical long-lived state,
contrary to a resonance. Nevertheless, the presence of an antibound or virtual state at
low energy has an important effect on the nucleon-nucleon cross section in the 1 S 0
channel for very small scattering energies. Indeed, when E → 0, the only important
channel is the 1 S 0 channel, as the 3 D 1 channel wave function disappears due to its
centrifugal barrier. Consequently, the knowledge of the effective range expansion is
sufficient to determine the behavior of nucleon-nucleon cross section at low energy.
As the scattering length is large and negative, one has to have an antibound state
of energy close to zero, which has been verified numerically. The presence of an
antibound state close to particle-emission threshold generates a more pronounced
localization of scattering states in the nuclear zone. Consequently, the scattering
cross section is much larger than in the case where no antibound state occurs.
The radial wave functions of the antibound state of dineutron and deuteron,
and virtual state of diproton, are depicted in Fig. 4.1. The antibound characters of
the dineutron and singlet deuteron states are immediately noticed as they increase
exponentially along the real r-axis (see Fig. 4.1). Their small energy, however,
makes the increase almost linear. The wave function of diproton virtual state has an
imaginary part comparable to that of dineutron and singlet deuteron states. Indeed,
the latter are purely imaginary, and all singlet systems have comparable energies.
However, due to its positive width, the diproton wave function possesses a nonzero
real part, of the same order of magnitude as its imaginary part already for small
values of radius. This is in contrast with narrow resonance states, where real part is
dominant and imaginary part is much smaller in the nuclear zone.
The 3 S 1 and 3 D 1 channel wave functions of the (bound) triplet state of a
deuteron are shown in Fig. 4.2. As is the case of all nucleon-nucleon interactions,
the 3 S 1 channel is dominant, while the 3 D 1 channel wave function is small but not
negligible. Its content has been calculated and amounts to 4.24%, close to the exact
value of 4.51% of the N 3 LO interaction [24], which is obtained by solving the twobody relative Schrödinger equation in momentum space. It has been checked that
the second maximum present in the 3 D 1 channel wave function, and to a lesser
extent in the 3 S 1 channel wave function, is not due to numerical inaccuracy, but
arises probably from the use of N 3 LO interaction. Indeed, this minimum has been
also noticed in Ref. [26]. The very smooth decrease of the bound 3 S 1 and 3 D 1
channel wave functions up to 25 fm indicates the completeness of Berggren basis
155
The energies and widths obtained for the unbound 0 + state of dineutron,
diproton, and deuteron are of particular interest, as they usually cannot be calculated
with standard methods. One can see that one obtains these unbound states at energy
close to zero, in accordance with experimental data. The real part of the energy of the
diproton is nearly the same as that of the dineutron or the excited T = 1 deuteron
state, whereas its imaginary part is close to 1 MeV in absolute value. This shows
that the Coulomb interaction present in the Hamiltonian can drastically change the
character of S-matrix pole of the two-body systems. In fact, the diproton ground
state is not a resonance, but a virtual state, as it has a negative energy and a positive
width.
An antibound or virtual state cannot be associated to a physical long-lived state,
contrary to a resonance. Nevertheless, the presence of an antibound or virtual state at
low energy has an important effect on the nucleon-nucleon cross section in the 1 S 0
channel for very small scattering energies. Indeed, when E → 0, the only important
channel is the 1 S 0 channel, as the 3 D 1 channel wave function disappears due to its
centrifugal barrier. Consequently, the knowledge of the effective range expansion is
sufficient to determine the behavior of nucleon-nucleon cross section at low energy.
As the scattering length is large and negative, one has to have an antibound state
of energy close to zero, which has been verified numerically. The presence of an
antibound state close to particle-emission threshold generates a more pronounced
localization of scattering states in the nuclear zone. Consequently, the scattering
cross section is much larger than in the case where no antibound state occurs.
The radial wave functions of the antibound state of dineutron and deuteron,
and virtual state of diproton, are depicted in Fig. 4.1. The antibound characters of
the dineutron and singlet deuteron states are immediately noticed as they increase
exponentially along the real r-axis (see Fig. 4.1). Their small energy, however,
makes the increase almost linear. The wave function of diproton virtual state has an
imaginary part comparable to that of dineutron and singlet deuteron states. Indeed,
the latter are purely imaginary, and all singlet systems have comparable energies.
However, due to its positive width, the diproton wave function possesses a nonzero
real part, of the same order of magnitude as its imaginary part already for small
values of radius. This is in contrast with narrow resonance states, where real part is
dominant and imaginary part is much smaller in the nuclear zone.
The 3 S 1 and 3 D 1 channel wave functions of the (bound) triplet state of a
deuteron are shown in Fig. 4.2. As is the case of all nucleon-nucleon interactions,
the 3 S 1 channel is dominant, while the 3 D 1 channel wave function is small but not
negligible. Its content has been calculated and amounts to 4.24%, close to the exact
value of 4.51% of the N 3 LO interaction [24], which is obtained by solving the twobody relative Schrödinger equation in momentum space. It has been checked that
the second maximum present in the 3 D 1 channel wave function, and to a lesser
extent in the 3 S 1 channel wave function, is not due to numerical inaccuracy, but
arises probably from the use of N 3 LO interaction. Indeed, this minimum has been
also noticed in Ref. [26]. The very smooth decrease of the bound 3 S 1 and 3 D 1
channel wave functions up to 25 fm indicates the completeness of Berggren basis