The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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3. The derivation of nuclear forces from chiral Lagrangians
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Figure 3.23: Leading contributions to the three-nucleon potential: irreducible twopion
exchange diagrams and irreducible one-pion exchange graph with the NN contact
interaction. Graphs which result from the interchange of the nucleon lines and
from the application of time reversal operation are not shown. In the case of diagram
9, one should sum over all possible time orderings. For remaining notations
see fig. 3.6.
in this case the same result within the projection formalism and time-ordered perturbation theory
and can use the Feynmann diagram technique, since no reducible topologies appear. Further, the
7r7r N N vertex contains a time derivative of the pion field, that has been counted as one power
of small momentum. But since the energy is conserved at each vertex when one calculates a
Feynmann diagram, this time derivative yields a difference of nucleon kinetic energies, which is of
higher order.
The first term in the third line of eq. (3.254) subsurnes the irreducible diagrams 1-8 in fig. 3.23,
representing time-ordered perturbation theory result for the two-pion exchange 3N force. Analogously,
the se co nd term in the first line of eq. (3.254) refers to the irreducible one-pion exchange
3N force involving contact interactions with four nucleon legs and without derivatives. The corresponding
graph is shown in fig. 3.23 (10). The diagrams depicted in fig. 3.23 define the complete
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