The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.9. Three-nucleon potential 115
leading-order 3N force within time-ordered perturbation theory. It is weIl known that the static
2 3 4
5 6
Figure 3.24: Leading contributions to the three-nucleon potential: reducible twopion
exchange diagrams and reducible one-pion exchange graphs with the NN contact
interaction. For notations see figs. 3.6, 3.23.
TPE 3N force cancels against the recoil corrections of the nucleons to the static OPE 2N potential,
when the latter is iterated in the Lippmann-Schwinger equation [191], [46]. The same sort
of cancelation happens for the leading TPE and OPE 3N forces related to diagrams 1-8 and 10
in fig. 3.23, as has been found by van KoIck [77]. Since the energy dependence of the NLO 2N
potential is entirely given by recoil corrections of the nucleons to the static OPE, one can describe
systems of three or more nucleons using the energy-independent part of the 2N potential and
without explicit three- and many-nucleon forces (at next-to-Ieading order in the chiral expansion
for the potential). However, such type of cancelations does not help to remove the problems
in the two-nucleon sector due to the explicit energy-dependence as noted before (i.e. that the
wavefunctions are only orthonormal to the order one is working).
Let us now take a closer look at the 3N force obtained with the projection formalism. As already
stated above, for the TPE graph 9 in fig. 3.23 one obtains the same result using both schemes.
Apart from the irreducible TPE diagrams 1-8 and OPE diagram 10 in fig. 3.23, one also has to
take into account "reducible" graphs shown in fig. 3.24. The diagrams 1-4 in this figure refer to the
last two terms in the third line of eq. (3.254), whereas the graphs 5 and 6 involving the N N contact
interactions arise from the terms in the fourth line of this equation. Obviously, the cancelation of
the leading 3N force with the iterated energy dependent part of the 2N potential, as observed in
the context of old-fashioned perturbation theory, is now not possible because the potential in the
projection formalism is energy independent. Nevertheless, the leading-order 3N force completely
vanishes in the method of unitary transformation, although the mechanism of the cancelation
is different from the above case. In the projection formalism, the 3N force vanishes because of