The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.9. Three-nuc1eon potential
2 3
Figure 3.26: Leading three-nuc1eon force with b,-isobar excitations. All possible
time orderings should be considered. For notations see figs. 3.6, 3.16 and fig. 3.23.
more-nuc1eon system. First steps in this direction have been done in refs. [118], [196]. In the
first work, Friar et al. consider the constraints on a possible form of the TPE force arising from
requirement of its (approximate) chiral symmetry. They pointed out that any TPE 3N force can
be expressed as
where
117
(3.338)
(3.339)
Here ij and ij' are the moment um transfers between the nuc1eons 1, 3 and 3, 2. Note that these
expressions are valid modulo l/m corrections. The authors of ref. [118] extracted the coefficients
a, b, C and d from various existing 3N forces, which are based on different models, and compared
them with the predictions obtained within the framework of chiral perturbation theory without
explicit b,'s. In the latter case, the leading order values of these coefficients can be calculated
from the 3N force shown in fig. 3.25 (1). The values of the relevant 7r7rNN couplings Cl , C3 and
C4 are known from the pion-nuc1eon sector, as discussed in sec. 3.8.2 and thus the coefficients a,
b, c and d can be predicted without free parameters. It turns out that although the signs of these
coefficients are always the same, their values differ substantially (by factors of two) for different
three-nuc1eon forces. Furthermore, to leading order in low-momentum expansion CHPT leads to
the largest absolute values for the coefficients a, b and d48 and to c = O. The c-term does not
vanish for the Tucson-Melbourne (TM) force [43] and thus violates chiral constraints. Therefore
in ref. [118] it is recommended to drop this c-term in the TM force.
In the second work, Hüber et al. analyse the effects of the 3N force corresponding to the second
graph in fig. (3.25) in elastic neutron-deuteron scattering at low-energies. They found that the
inclusion of such a short-range-long-range force might considerably improve the agreement for
the analyzing power Ay• This study is, however, merely qualitative, since only a part of the
leading-order 3N force resulting from CHPT has been included and no attempt was made to
obtain a reasonable fit to data.
An important observation has been done by van Kolck in ref. [77]. He pointed out that several
vertices entering the expressions for the leading 3N force are saturated by virtual b,-isobar ex-
4 8 Note, however, that the authors of ref. [118) used the values for Cl,3,4 as given in [71) that are somewhat larger
than the ones from the later publication [195], which we adopt in this work.