The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

quark models one typically represents nucleonic degrees of freedom in terms of 3-quark states.
This allows to fit simply the quantum numbers of the nucleons as well as to provide their color
neutrality. The property of confinement which is an important phenomenon of QCD is usually
simulated by some phenomenological confining potentials [55].
Recently a combined model of the nuclear force, the so-called Moscow potential, has been proposed
[56]. This can be considered as furt her extension and improvement of earlier models of the
combined type, which have been developed by Faessler et al. [57], [58]. There the interactions
at large separations are described in terms of meson-exchanges. At short distances one notes
that the six-quark state no longer resembles two distinct nucleons (Le. two three-quark nucleon
clusters), as it happens for large separations. Such considerations lead to a suppression of the local
repulsive co re of the nuclear force, which is an obligatory attribute of boson-exchange models.5
The effects of the repulsive core are simulated by additional deeply-lying bound states, which are
generated by the Moscow-type potentials. As a consequence, additional internal nodes appear
in the deuteron and scattering wave functions. Furthermore, such a suppression of the repulsive
co re leads to a sm aller value for the wN N coupling constant compared to boson-exchange models,
which is consistent with the one predicted from SU(3)-symmetry. The suppression of the repulsive
core in quark model calculations was also pointed out earlier by Harvey [59].
A more systematic attempt to include information from QCD is based on effective field theories
(EFT). Already 20 years aga Weinberg [60] pointed out that requiring the (approximate) SU(2) x
SU(2) chiral symmetry, which is evident from the QCD Lagrangian, leads to a model-independent
and systematic low-energy expansion in momenta for the S-matrix. Chiral symmetry of the QCD
Lagrangian is not a symmetry of the physical vacuum, which is only invariant under a smaller
SU(2) subgroup. Thus, chiral symmetry is spontaneously broken, which can also be verified
from the hadronic spectrum. As a consequence, according to Goldstone's theorem [61], [62], one
observes three light pseudoscalar bosons, which would be massless in the exact chiral limit and can
be identified with pions. Weinberg illustrated his idea with a systematic low-energy expansion
of the amplitude on the example of 11'11' scattering. Such an expansion is possible, since in the
exact chiral limit only derivative interactions between pions are allowed. Thus, for vanishing
momenta pions become free particles. The original idea of Weinberg has been worked out in
detail in calculations by Gasser and Leutwyler for 11'11' scattering and many other processes [63],
[64]. For a review article see ref. [65] . Once the coupling constants of mesonic interactions in
the most general chiral invariant Lagrangian are fixed from some processes, various predictions
can be made for other reactions and observables, allowing for non-trivial tests of the Standard
Model. Even two-loop calculations have been performed recently [66]. Further, external fields can
be incorporated quite naturally and systematically. This opens the possibility to study various
processes including photons.
It is well-known how to couple non-Goldstone degrees of freedom in a chiral invariant manner
[67]. In 1988 the technique of the effective Lagrangian has been applied to calculate various pionnucleon
amplitudes to one-loop [68]. In this approach, the relativistic treatment of nucleon fields
in loops caused an intrinsic problem with chiral power counting. This is because an additional
scale, namely the nucleon mass, which remains finite in the chiral limit is introduced due to
baryon propagators. The problem was successfully solved within the heavy baryon formulation
of chiral perturbation theory (CHPT) [69], [70]. The basic idea of this scheme is to integrate
out the "heavy" component of the baryon field. This can be carried out explicitly in a Lorentz
covariant way in terms of velocity-dependent fields. This allowed numerous applications for the
5The local repulsive core results in the boson-exchange models from exchanges of vector wand p mesons.
5