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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96<br />
_ . . -.:;: " :,,,_ ._ . . _.<br />
3. <strong>The</strong> derivation of nuc1ear forces from chiral Lagrangians<br />
_. . _ .:: . .,. :",_. _. . _ .<br />
_ . . _._. __ ._ . . _.<br />
_ . . _._."._._ . . _.<br />
2<br />
- . . _._. __ ._ . . -.<br />
_ . . _._.04._._ . . _.<br />
_. __ . :: ,., 1""_ ._ . . _.<br />
Figure 3.12: Reducible graphs. In (1), a truly reducible diagram is shown. In the<br />
projection formalism, one has graphs like (2) and (3). <strong>The</strong>se correspond to diagram<br />
9 <strong>in</strong> fig. 3.9. <strong>The</strong> horizontal dot-dashed l<strong>in</strong>es count the free energies of the particles<br />
cut.<br />
respectively. <strong>The</strong> lead<strong>in</strong>g order potential can then be written as:<br />
2 3 4 5 6<br />
Figure 3.13: NLÜ one-loop corrections to the four-nucleon contact <strong>in</strong>teractions <strong>in</strong><br />
the projection formalism. For notations see fig. 3.6.<br />
3<br />
(3.269)<br />
<strong>The</strong> first corrections to this result appear at order two. We now enumerate all such corrections.<br />
S<strong>in</strong>ce we want to compare both schemes, we will first discuss the results obta<strong>in</strong>ed with<strong>in</strong> oldfashioned<br />
perturbation theory and then present the potential derived us<strong>in</strong>g the method of unitary<br />
transformation. We start with the tree-diagram contribution given by the graph of fig. 3.8 with<br />
various contact <strong>in</strong>teractions from eq. (3.237). This leads to the potential<br />
(2) VNN,tree C �2 C k2 (C �2 � �<br />
C k2 )(<br />
1 q + 2 + 3 q + 4 0"1 • 0"2 + Z 5 . x q)<br />
2<br />
+ C6 (if· o\) (if· i'h) + C7 (k . ih)(k . (2 ) , (3.270)<br />
) ' C 0\ + ih (k ;;'I