The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

116 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
2 3
Figure 3.25: Three-nucleon force: TPE, OPE and contact interaction. In the case
of diagrams 1 and 2, all possible time orderings should be considered. For notations
see figs. 3.6, 3.23 and fig. 3.14.
an exact cancelation between the contributions from the "reducible" and irreducible OPE and
TPE diagrams. This cancelation was recently pointed out in [185] for the case of an expansion
in the pion-nucleon coupling constant and adopting the static approximation for nucleons. To
understand why the leading 3N force vanishes, let us take a closer look at the terms in the third
line of eq. (3.254). The contributions from the irreducible TPE diagrams 1-8 in fig. 3.23 can be
expressed schematically as:
(3.336)
where we pulled out the common factor M representing the spin, isospin and momentum structure,
which is obviously the same for all graphs 1-8 in fig. 3.23. The contribution from the "reducible"
diagrams 1-4 in fig. 3.24 can be expressed as
[ 2 2 ] Wl + W2
+ -2- w1 w2 WIW2
--2 M=2 w1 22M. w2
(3.337)
The cancelation is now evident. The same sort of cancelation can be observed for the irreducible
diagram 10 in fig. 3.23 and the "reducible" graphs 5 and 6 in fig. 3.24 involving contact interactions.
We conclude that there is no three-nucleon force at the order v = -l.
Let us now consider the 3N force at order v = 0 related to the effective potential (3.255). The
terms in the second line of eq. (3.255) refer to the two-pion exchange diagram 1 in fig. 3.25. The
first term in this equation corresponds to the contact force shown in fig. 3.25 (3). Finally, the
one pion exchange graph 2 involving contact interactions with four nucleon legs arises from the
second and third terms in eq. (3.255). In all cases the method of unitary transformation does not
introduce any new aspects, since it yields the same result as time-ordered perturbation theory.
Within the last approach, this 3N force has been calculated and discussed by van Kolck [77].
In that reference a complete expression for the leading chiral 3N force is given. Unfortunately,
the inclusion of the leading 3N interaction intro duces several new parameters. While the 1f N N
couplings D1 and D2 in eq. (3.236) can, in principle, be determined from the processes like 1fdeuteron
scattering or 1f production and absorption on N N system, the remaining three contact
interactions El,2,3 can only be fixed from data involving systems with more than two nucleons.
At present, this 3N force has not yet been applied in systematic calculations of the three- and