The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

176 B. Operators contributing to eq. (3.206) at order r
(a) i = 1,2
It can be seen from the inequalities (AA), (A.6) that in this case K, � 2. Using the
inequality i - j :S 2 we get from eq. (B.I)
l:Sr-2 .
(b) i = 3
The general inequality (A.5) leads immediately to
4. q � k + 2
1. j=O : l:Sr ,
11. j � 1 : l:S r - 1
(B.6)
(B.7)
Using eq. (A.6) with the number of pions p given by p = 4q + j - 4k - i we obtain from
eq. (B.I)
l :S r - 6(q - k) + 2 + 2i - 2j :S r - 6(q - k) + 8 :S r - 4 . (B.8)
The case i = 0 can be considered analogously:
1. q:Sk-2=?l:Sr-2,
2. q = k - 1 .
(a) j � 1
Putting p = 4 - j in eq. (AA) we see from eq. (B.I), that
l:Sr-2.
(b) j = 0
In this case we have K, � 2, as it follows from eq. (A.6). We obtain
3. q = k
It follows from eqs. (A.3) and (AA), that
l :S r .
l