The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

92 3. The derivation of nuc1ear forces from chiral Lagrangians
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Figure 3.6: Some of the one- and zero-partide processes that will not be considered.
Solid and dashed lines are nudeons and pions, respectively. The heavy dots denote
vertices from eqs. (3.231)-(3.238) with ßi = O.
5
I
(3.265)
Using the decompositions (3.258), (3.262) of the field operators one can express the interaction
Hamiltonian that corresponds to the density (3.231)-(3.238) in terms of creation and destruction
operators. The calculation of the effective potential (3.253)-(3.255) is performed at a fixed time
t = 0, at which the Heisenberg, Dirac (or interaction) and Schrödinger pictures coincide. To
obtain the final result one has to proceed in a way similar to time-ordered perturbation theory,
see, for example, [187]. The only difference is in the energy denominators and in the coefficients of
the operators entering eqs. (3.253)-(3.255). A detailed calculation of the effective potential within
a time-ordered perturbation theory as wen as the relevant matrix elements of the Hamiltonian
(3.231)-(3.238) can be found in the reference [161].
Figure 3.7: Leading order (LO) contributions to the NN potential: one-pion exchange
and contact diagrams. Graphs which result from the interchange of the two
nudeon lines are not shown. For notations see fig. 3.6.
Let us take a dos er look at various terms entering the expressions (3.253)-(3.255), before we
will give the explicit result for the effective potential. For that, we will use the diagrammatic
technique. It should be kept in mi nd that the graphs we will show below only represent the
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