The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two�nucleon potential
2 3
Figure 3.21: First corrections to the NN potential with Ll�excitations: One�loop
diagrams without intermediate pions. For notations see figs. 3.6, 3.16.
eq. (3.323) and represent the correction obtained with time�ordered perturbation theory. In
the projection formalism one has to take into account also "reducible" diagrams 5�8 in fig. 3.18
resulting from the last two terms in the second line of eq. (3.323). Note that all diagrams containing
vertex corrections with one 1r1r N Ll vertex give no contributions for the same reason as in the case
without Ll.
There are many new vertex and self�energy corrections with the intermediate Ll 's that contain
contact interactions. In fig. 3.19 we show the dass of irreducible diagrams that correspond to the
second term in the first line and to the second and third terms in the last line of eq. (3.323). This
is the complete contribution in time�ordered perturbation theory. In the projection formalism one
has an additional correction resulting from two "reducible" graphs of fig. 3.20, which are related
to the terms in the third line of eq. (3.323).
, /
, /
,
,
'( '(
/
/
,
/
,
/
/
/
/
,
/
2 3 4 5
Figure 3.22: Leading two�pion exchange contributions with single and double Ll�
excitations to the effective potential. For notations see fig. 3.6, 3.16.
A completely new type of the one�loop diagrams with only contact interactions is related to the
first operator in the last line of eq. (3.323). The three graphs with single and double Ll�excitations
are shown in fig. 3.21. Note that these diagrams yield a purely short�range contribution that
renormalizes the N N contact interactions without derivatives. This is because no momentum
dependence is introduced by the corresponding vertices and energy denominators. The only
,
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