4.2 Study of Two-Nucleon Systems with Realistic Interactions
151
deuteron. Even though both systems are unbound, the dineutron as well as the 1 S 0
singlet state of the deuteron are antibound states of energy close to particle-emission
threshold, whereas the S-matrix pole of the diproton has a low negative energy and
a positive width. Thus, diproton, dineutron, and the 1 S 0 singlet state of the deuteron
are not resonances. Nevertheless, as they lie close to the real energy axis, their
indirect effect on cross sections is experimentally visible. In fact, np cross section
at zero energy is close to 20 barns [4].
While the Lagrangian of quantum chromodynamics is well known [5–8], the
nucleon-nucleon interaction cannot be directly calculated from quantum chromodynamics. This arises from the non-perturbative nature of quantum chromodynamics
in the low energy region. In order to counteract this situation, Lagrangian of
the quantum chromodynamics expressed in terms of quark and gluon fields is
replaced by an effective interaction whose degrees of freedom are nucleons and
mesons. Recent versions of this interaction bear the same chiral symmetry as the
initial Lagrangian of quantum chromodynamics. Moreover, the interaction can be
calculated perturbatively in the effective field theory which can be stated in one
theorem [9, 10]:
For a given set of asymptotic states, perturbation theory with the most general
Lagrangian containing all terms allowed by the assumed symmetries will yield the
most general S-matrix elements consistent with analyticity, perturbative unitarity,
cluster decomposition and the assumed symmetries.
Effective field theory [9–15] seems to be an attractive approach to describe
physics of nucleons and nuclei at low energies, i.e., at energies e below a certain
energy scale Λ imposed by the properties of a studied system and observables one
wish to describe. This scale could be the nucleon mass or the pion mass in the chiral
effective field theory, or the core excitation energy in halo effective field theory.
In the effective field theory, relevant degrees of freedom at e Λ are explicitly
taken into account whereas other degrees of freedom, related to energies e ≥ Λ,
are taken into account effectively in the renormalized coupling constants. If the
interaction parameters converge to a finite value when Λ → +∞, the interaction
is deemed as renormalizable. In the low-energy domain, the interactions stemming
from the chiral effective theory are written as a perturbative expansion in a small
parameter of the theory, such as e/Λ. In principle, effective field theory gives rise to
an infinite number of terms in the e/Λ expansion. For a fixed order in the expansion,
the number of couplings is finite and one can renormalize the obtained interaction
order by order.
Effective field theory can be applied to nuclei at low energy, where e ≤
300 MeV typically, whereas the nucleon mass M is much heavier, as M ∼ 1 GeV.
Consequently, it is possible to devise quickly converging power series expansions in
terms of e/M ratios. The high energy degrees of freedom in this theory are indeed
integrated out in order to suppress in particular the hard core nucleon-nucleon
interaction which scatters nucleons away from the shell model space and is difficult
to treat numerically. The low-momentum interactions obtained in this way can be
conveniently applied in various basis expansion methods to solve the Schrödinger
151
deuteron. Even though both systems are unbound, the dineutron as well as the 1 S 0
singlet state of the deuteron are antibound states of energy close to particle-emission
threshold, whereas the S-matrix pole of the diproton has a low negative energy and
a positive width. Thus, diproton, dineutron, and the 1 S 0 singlet state of the deuteron
are not resonances. Nevertheless, as they lie close to the real energy axis, their
indirect effect on cross sections is experimentally visible. In fact, np cross section
at zero energy is close to 20 barns [4].
While the Lagrangian of quantum chromodynamics is well known [5–8], the
nucleon-nucleon interaction cannot be directly calculated from quantum chromodynamics. This arises from the non-perturbative nature of quantum chromodynamics
in the low energy region. In order to counteract this situation, Lagrangian of
the quantum chromodynamics expressed in terms of quark and gluon fields is
replaced by an effective interaction whose degrees of freedom are nucleons and
mesons. Recent versions of this interaction bear the same chiral symmetry as the
initial Lagrangian of quantum chromodynamics. Moreover, the interaction can be
calculated perturbatively in the effective field theory which can be stated in one
theorem [9, 10]:
For a given set of asymptotic states, perturbation theory with the most general
Lagrangian containing all terms allowed by the assumed symmetries will yield the
most general S-matrix elements consistent with analyticity, perturbative unitarity,
cluster decomposition and the assumed symmetries.
Effective field theory [9–15] seems to be an attractive approach to describe
physics of nucleons and nuclei at low energies, i.e., at energies e below a certain
energy scale Λ imposed by the properties of a studied system and observables one
wish to describe. This scale could be the nucleon mass or the pion mass in the chiral
effective field theory, or the core excitation energy in halo effective field theory.
In the effective field theory, relevant degrees of freedom at e Λ are explicitly
taken into account whereas other degrees of freedom, related to energies e ≥ Λ,
are taken into account effectively in the renormalized coupling constants. If the
interaction parameters converge to a finite value when Λ → +∞, the interaction
is deemed as renormalizable. In the low-energy domain, the interactions stemming
from the chiral effective theory are written as a perturbative expansion in a small
parameter of the theory, such as e/Λ. In principle, effective field theory gives rise to
an infinite number of terms in the e/Λ expansion. For a fixed order in the expansion,
the number of couplings is finite and one can renormalize the obtained interaction
order by order.
Effective field theory can be applied to nuclei at low energy, where e ≤
300 MeV typically, whereas the nucleon mass M is much heavier, as M ∼ 1 GeV.
Consequently, it is possible to devise quickly converging power series expansions in
terms of e/M ratios. The high energy degrees of freedom in this theory are indeed
integrated out in order to suppress in particular the hard core nucleon-nucleon
interaction which scatters nucleons away from the shell model space and is difficult
to treat numerically. The low-momentum interactions obtained in this way can be
conveniently applied in various basis expansion methods to solve the Schrödinger