The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

which is also based on the one-boson exchange model [29]. This potential also takes into account
charge dependence of the nuclear force.
Also, a more phenomenological approach of the Argonne group to keep in the potential explicitly
only the one-pion exchange and to represent all remaining contributions in a general operator form
led after fitting of 40 adjustable parameters to a quite accurate description of the two-nucleon
scattering data with an excellent X 2 per datum of 1.09 [37].
All these boson-exchange models appear to be very successful in describing the two-nucleon
scattering data as weIl as the deuteron properties. In spite of this success in the pure twonucleon
sector there exist so me situations in which the OBE models do not allow far satisfactory
explanations and description of data. Since usually the two-nucleon scattering data are used to fix
the free parameters in the potentials, only on-energy-shell physics is reproduced correctly. Offenergy-shell
effects can only be tested in reactions with more than two nucleons and in processes
with external probes.3 Consequently, due to this off-shell ambiguity, the value of, for instance,
the triton binding energy varies remarkably when calculations are performed with the different
two-nucleon forces. None of the existing so-called realistic potentials lead to a correct value of
the triton binding energy. Typically, one observes underbinding of about 5-10%, which can be
explained in terms of a missing three-nucleon force (3NF). Indeed, the inclusion of a three-body
force allows to describe the triton binding energy correctly. One should point out that up to
now much less is known about the nature of the three-body forces compared to the two-body
interactions. This has partly historical reasons: only relatively recently it became possible to
solve the three-body Faddeev equations exactly for any type of realistic two-body force, after all
necessary technical and computational tools were worked out and sufficient computer powers were
available [38], [39], [41]. In addition to these technical and computational difficulties, the effects
of the three-body force in most cases are small and require precise experimental measurements of
the observables.
At presence several models for the three-body force are available. Some of them like the Fujita­
Miyazawa [42] or the Tucson-Melbourne [43] forces are based on the two-pion exchange with one
intermediate ß excitation. Such a two-pion exchange interaction represents the longest range
part of the 3NF. Another model proposed by the Brazil group includes in addition 7f-P and p-p
exchanges [44]. The Urbana-Argonne group has worked out a purely phenomenological 3NF [45].
Finally, the Tucson-Melbourne force has been extended to take into account also the 7f-P and
p-p exchanges (the so-called Tucson-Melbourne model) [46], [47]. For various combinations of
the two- and three-nucleon interactions one can always adjust parameters (typically the values of
the cut-off in the 3NF) to exactly reproduce the triton binding energy [40]. For some observables
the inclusion of the 3NF does, however, not lead to an improved description compared to the
calculations with the purely two-body interactions [41]. The most prominent example of such an
observable is an analyzing power Ay in elastic nd scattering at low energies. The purely twonucleon
calculations yield results which are about 25% off the experimental values. Inclusion of
3 To avoid misunderstanding we note that, in general, only on-energy-shell effects can be observed. One should
understand the "off-energy-shell effects" in this context as follows. Assume, one has some definite two- and morebody
forces. Then, one can always unitarily transform the full Hamiltonian and obtain new two- and more-body
interactions. The old and new two-body potentials are phase-equivalent and give the same two-body S-matrix.
However, to calculate three- and more-body observables one needs the off-shell two-body T-matrix, which is, in
general, modified after performing the unitary transformation. This is precisely what we understand under such
"off-energy-shell effects" . The differences in the off-shell T-matrices are compensated by the modified many-body
interaction and the final result for on-shell quantities is, clearly, the same before and after performing the unitary
transformation.
3