The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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