The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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3.8. Two�nucleon potential<br />
2 3<br />
Figure 3.21: First corrections to the NN potential with Ll�excitations: One�loop<br />
diagrams without <strong>in</strong>termediate pions. For notations see figs. 3.6, 3.16.<br />
eq. (3.323) and represent the correction obta<strong>in</strong>ed with time�ordered perturbation theory. In<br />
the projection formalism one has to take <strong>in</strong>to account also "reducible" diagrams 5�8 <strong>in</strong> fig. 3.18<br />
result<strong>in</strong>g from the last two terms <strong>in</strong> the second l<strong>in</strong>e of eq. (3.323). Note that all diagrams conta<strong>in</strong><strong>in</strong>g<br />
vertex corrections with one 1r1r N Ll vertex give no contributions for the same reason as <strong>in</strong> the case<br />
without Ll.<br />
<strong>The</strong>re are many new vertex and self�energy corrections with the <strong>in</strong>termediate Ll 's that conta<strong>in</strong><br />
contact <strong>in</strong>teractions. In fig. 3.19 we show the dass of irreducible diagrams that correspond to the<br />
second term <strong>in</strong> the first l<strong>in</strong>e and to the second and third terms <strong>in</strong> the last l<strong>in</strong>e of eq. (3.323). This<br />
is the complete contribution <strong>in</strong> time�ordered perturbation theory. In the projection formalism one<br />
has an additional correction result<strong>in</strong>g from two "reducible" graphs of fig. 3.20, which are related<br />
to the terms <strong>in</strong> the third l<strong>in</strong>e of eq. (3.323).<br />
, /<br />
, /<br />
,<br />
,<br />
'( '(<br />
/<br />
/<br />
,<br />
/<br />
,<br />
/<br />
/<br />
/<br />
/<br />
,<br />
/<br />
2 3 4 5<br />
Figure 3.22: Lead<strong>in</strong>g two�pion exchange contributions with s<strong>in</strong>gle and double Ll�<br />
excitations to the effective potential. For notations see fig. 3.6, 3.16.<br />
A completely new type of the one�loop diagrams with only contact <strong>in</strong>teractions is related to the<br />
first operator <strong>in</strong> the last l<strong>in</strong>e of eq. (3.323). <strong>The</strong> three graphs with s<strong>in</strong>gle and double Ll�excitations<br />
are shown <strong>in</strong> fig. 3.21. Note that these diagrams yield a purely short�range contribution that<br />
renormalizes the N N contact <strong>in</strong>teractions without derivatives. This is because no momentum<br />
dependence is <strong>in</strong>troduced by the correspond<strong>in</strong>g vertices and energy denom<strong>in</strong>ators. <strong>The</strong> only<br />
,<br />
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