The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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