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 3.9: First corrections to the NN potential in the projection formalism: twopion
exchange diagrams. For notations see fig. 3.6.
(1), one can e.g. express diagram 9 of fig. 3.9 as a sum of two graphs as depicted in fig. 3.12
(2),(3). All diagrams involving contact interactions, shown in fig. 3.13, follow from the fourth line
and, as already noted above, from the second term in the first line of eq. (3.254). We remark
that the three terms in the fifth line add up to zero. We have nevertheless made them explicit
here since in time-ordered perturbation theory, these terms are treated differently and lead to the
recoil correction, i.e. the explicit energy-dependence, of the two-nucleon potential. The last term
in eq. (3.254) corresponds to the vacuum fluctuation diagram shown in fig. 3.6 and will not be
discussed below.
Let us now consider the next-to-next-to-leading order corrections (NNLO) represented by the
equation (3.255). The corresponding diagrams have the same structure as the graphs 1-4 in
fig. (3.9) except that now one 7r7rNN vertex contains two derivatives (or two pion mass insertions),
i. e. one more than the diagrams in fig. (3.9). The two last terms in the first line of eq. (3.255)
are referred to the graphs 4 and 5 in fig. 3.15. The diagrams 1-3 in this figure represent the
terms in the second line of eq. (3.255). The corresponding vertex correction diagrams are shown
in fig. 3.14.
We will now give explicit expressions for the N N potential. As already noted ab ove , its leading
part is given by just one pion exchange with both vertices coming from eq. (3.231) plus contact
interactions corresponding to eq. (3.234). As pointed out in ref. [78] in the context oftime-ordered
perturbation theory, one has to take the static limit for the nucleons in the one-pion exchange
term, since the recoil and relativistic corrections lead to higher order contributions. The same
holds true in the method of unitary transformation, see eq. (3.253). Thus, both methods give
the same result for the leading order potential. In what follows, we will use the slightly different
notation from refs. [78], [127]. In particular, the spin and isospin matrices a and T satisfy the
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