The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

110 3. The derivation of nuclear forces from chiral Lagrangians
find the following expressions:
V (6 -3N )
elf
V(8-3N)
elf
(3.323)
We will now discuss these expressions in more detail. Obviously, the leading order contribution
Figure 3.20: First corrections to the NN potential with ß-excitations: reducible
self-energy corrections to the four-nucleon contact interactions. For notations see
figs. 3.6, 3.16.
(3.322) to the effective N N potential is not changed by the explicit inclusion of the ß.47 At
NLO one has many additional contributions to the two-pion exchange diagrams and to the vertex
corrections and self-energy graphs. The first four lines of eq. (3.323) give formally the complete
NLO potential in the case without ß. In addition to the one-pion exchange diagrams containing
self-energy insertions and vertex corrections discussed in sec. 3.8.1 and shown in figs. 3.10, 3.11
one has to take into account the graphs in figs. 3.17, 3.18. In the diagrams shown in fig. 3.17,
which are related to the first term in the second line and to the last term in eq. (3.323), no purely
nucleonic intermediate states appear if one sums up over all possible time orderings. Therefore, the
projection formalism yields the same result as time-ordered perturbation theory, which agrees after
summation over all time orderings with the one obtained using the Feynmann diagram technique.
Again one should keep in mind that this statement holds true because of the static approximation
for the nucleons. Note that we have depicted in fig. 3.17 the corresponding Feynmann diagrams
without showing all time orderings. The one-pion exchange diagrams of fig. 3.18 have a different
topology. The irreducible graphs 1-4 correspond again to the first term in the second line of
47 Note that there are new contributions to the one-nucleon operators due to the diagram with an intermediate
delta-excitation, which will not be considered in what folIows.