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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110 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians<br />
f<strong>in</strong>d the follow<strong>in</strong>g expressions:<br />
V (6 -3N )<br />
elf<br />
V(8-3N)<br />
elf<br />
(3.323)<br />
We will now discuss these expressions <strong>in</strong> more detail. Obviously, the lead<strong>in</strong>g order contribution<br />
Figure 3.20: First corrections to the NN potential with ß-excitations: reducible<br />
self-energy corrections to the four-nucleon contact <strong>in</strong>teractions. For notations see<br />
figs. 3.6, 3.16.<br />
(3.322) to the effective N N potential is not changed by the explicit <strong>in</strong>clusion of the ß.47 At<br />
NLO one has many additional contributions to the two-pion exchange diagrams and to the vertex<br />
corrections and self-energy graphs. <strong>The</strong> first four l<strong>in</strong>es of eq. (3.323) give formally the complete<br />
NLO potential <strong>in</strong> the case without ß. In addition to the one-pion exchange diagrams conta<strong>in</strong><strong>in</strong>g<br />
self-energy <strong>in</strong>sertions and vertex corrections discussed <strong>in</strong> sec. 3.8.1 and shown <strong>in</strong> figs. 3.10, 3.11<br />
one has to take <strong>in</strong>to account the graphs <strong>in</strong> figs. 3.17, 3.18. In the diagrams shown <strong>in</strong> fig. 3.17,<br />
which are related to the first term <strong>in</strong> the second l<strong>in</strong>e and to the last term <strong>in</strong> eq. (3.323), no purely<br />
nucleonic <strong>in</strong>termediate states appear if one sums up over all possible time order<strong>in</strong>gs. <strong>The</strong>refore, the<br />
projection formalism yields the same result as time-ordered perturbation theory, which agrees after<br />
summation over all time order<strong>in</strong>gs with the one obta<strong>in</strong>ed us<strong>in</strong>g the Feynmann diagram technique.<br />
Aga<strong>in</strong> one should keep <strong>in</strong> m<strong>in</strong>d that this statement holds true because of the static approximation<br />
for the nucleons. Note that we have depicted <strong>in</strong> fig. 3.17 the correspond<strong>in</strong>g Feynmann diagrams<br />
without show<strong>in</strong>g all time order<strong>in</strong>gs. <strong>The</strong> one-pion exchange diagrams of fig. 3.18 have a different<br />
topology. <strong>The</strong> irreducible graphs 1-4 correspond aga<strong>in</strong> to the first term <strong>in</strong> the second l<strong>in</strong>e of<br />
47 Note that there are new contributions to the one-nucleon operators due to the diagram with an <strong>in</strong>termediate<br />
delta-excitation, which will not be considered <strong>in</strong> what folIows.