The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

108 3. The derivation of nuclear forces from chiral Lagrangians
A and B are valid also in this case. Furthermore, one has to include states contained in A ° with
.6.'s and without pions. This is discussed in appendix C. Let us now take a closer look at these
modifications.
\ /
\
\
\
,
,
,
\ I I /
, /
\
I
\
\
/
/
/
/
/
/
,
,
\ ,
I
2 3 4
\ \
, " I I
, "
, " "
\ \
I I
5 6 7 !\
Figure 3.18: First corrections to the NN potential with .6.-excitations: irreducible
(1-4) and reducible (5-8) vertex and self-energy corrections to the one-pion exchange.
For notations see figs. 3.6, 3.16.
For the projected operator Aa A1] we make the same ansatz as in the case without .6., namely that
it consists of an equal number of vertices and energy denominators. Since the inclusion of the .6.
does not affect the power counting scheme, equations (3.210) and (3.212) are not modified. Here
and in what follows, we will denote by N the baryon number. Consequently, the minimal value
of VA is again given by eq. (3.213). Furthermore, min(vA) = 2 for the operator AOA1], as follows
from eq. (3.210). Therefore, we will again define the order l of the operator Aa A1] via eq. (3.214).
As discussed in appendix C, the inclusion of the ß's does not change the minimal value of the
chiral power v for the projected decoupling equation (3.206). Therefore, we can apply eqs.
(3.216)
(3.215),
without any changes. Thus, the system of equations (3.205) can be solved perturbatively
precisely in the same way as in the case without ß. The only modification is that at each order
r one has to start by solving the equation (3.217) projected with AO • It is shown in appendix C
that all operators Aa A/1] of the right-hand side of this equation are of higher orders l 2: r + 2.
The resulting operator A ° Ar 1] is then used to solve the remaining equations at order r projected
onto the states with pions.
Since equation (3.217) with 4k + i = 0 vanishes at orders r = 0 and r = 1, the starting equations
are again (3.221), (3.222). Therefore, the ansatz about the structure of the operator A can be
justified in the same way as in sec. 3.6. Finally, all expressions (3.225)-(3.229) can be applied
without any changes to obtain the effective potential.
We are now in the position to calculate the effective potential including .6.-excitations. We will
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