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-nuc1eon potential 107<br />
where i, j, k, 1 = 1 for (p3/2)J-tv, Us<strong>in</strong>g these projection operators one can decompose the Rarita<br />
Schw<strong>in</strong>ger field \[! J-t <strong>in</strong>to one sp<strong>in</strong> 3/2 and two sp<strong>in</strong> 1/2 components. To perform the nonrelativistic<br />
reduction one can proceed <strong>in</strong> the same way as for the pion-nucleon Lagrangian. One factors out<br />
the exponential factor exp (imv . x) from the nucleon and delta fields, where m is the nucleon mass,<br />
and <strong>in</strong>troduces the the small and large component fields via eq. (3.160). This leads to altogether<br />
six components for the field \[! J.t' Only the large component TJ-t of the sp<strong>in</strong> 3/2 projection of the<br />
Rarita-Schw<strong>in</strong>ger sp<strong>in</strong>or \[! J-t corresponds to the light field with the mass ß = mf::, - m. <strong>The</strong><br />
large mass scale m enters the free equations of motion for the rema<strong>in</strong><strong>in</strong>g five components. Thus,<br />
all these fields can be <strong>in</strong>tegrated out, which leads to local l/m corrections to the Lagrangian.<br />
Proceed<strong>in</strong>g <strong>in</strong> this way one ends up with the free Lagrangian (<strong>in</strong> the rest frame)<br />
where i, j = 1,2,3<br />
( 'l1 ,, )s:ij TV<br />
Lf::, O - - i Zuo - L.l. u gJ-tv j , (3.316)<br />
r _ TJ-tt<br />
are the isosp<strong>in</strong> <strong>in</strong>dices.46 <strong>The</strong> projection formalism <strong>in</strong>troduced <strong>in</strong> sections<br />
3.5, 3.6 can be generalized to <strong>in</strong>corporate the .6.'s. <strong>The</strong> subspace Ai of the full Fock space is<br />
now enlarged to <strong>in</strong>clude the states with any number of ß's apart from the nucleons and i pions.<br />
Different to the pure pion-nucleon <strong>in</strong>teractions, one also has to <strong>in</strong>clude the projection A 0, that<br />
conta<strong>in</strong>s any positive number of .6.'s but no pions. This requires an appropriate generalization<br />
of the projection formalism. We first note that the structure of the <strong>in</strong>teractions with the .6. 's <strong>in</strong><br />
\<br />
I<br />
\<br />
\<br />
- - -<br />
T<br />
5<br />
\<br />
I<br />
2<br />
\<br />
I<br />
\<br />
I<br />
3 4<br />
\<br />
\ \<br />
- - -<br />
- - -<br />
T T<br />
6 7<br />
Figure 3.17: First corrections to the NN potential with .6.-excitations: vertex corrections<br />
to the one-pion exchange. A class of diagrams is shown for which no purely<br />
nucleonic <strong>in</strong>termediate states are possible. For notations see figs. 3.6, 3.16.<br />
the effective Lagrangian is the same as for the correspond<strong>in</strong>g nucleonic vertices. <strong>The</strong> sp<strong>in</strong>-isosp<strong>in</strong><br />
structure is, of course, different but this does not affect the power count<strong>in</strong>g arguments, for which<br />
only the number of derivatives is of relevance. <strong>The</strong>refore, all results and conclusions of appendices<br />
46 <strong>The</strong> isosp<strong>in</strong> 3/2 field T is described by three isosp<strong>in</strong> doublets T l , T 2 , T 3 that satisfy the condition r i T i (x) = O.