The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

derivative. Further helpful equalities are
1);2:1 ,
1);2:-2+p
173
(A.5)
(A.6)
Whereas (A.5) is generally valid since no renormalizable interactions are allowed by chiral symmetry,
(A.6) follows from simple inspection of eq. (3.211): for n = 0 there are at least two derivatives
and for n 2: 2 it is immediately apparent.
Below we will examine the chiral power v for all terms entering the left-hand side of eq. (3.206).
• )..4k +iHK'fl
v = 4 - 3N - 4k - i + I); • (A.7)
This follows from eq. (3.210) if one notes, that now there is one energy denominator less.
We then obtain from eqs. (A.3), (A.4), (A.6):
v 2: 4 - 3N, when k = 0 ,
v 2: 2 - 3N, when k > 0
• )..4k +i HK)..4Q+ j Az'fl
v = 4 - 3N - 4k - i + 2q + j + I); + l .
The following possibilities are to be considered:
(A.8)
1. q::;k-l
In this case the number of pions at the vertex is restricted by p 2: 4k + i - 4q - j.
Therefore, the inequality (A.6) takes the form
It then follows from eq. (A.8), that
2. q = k
Here we have:
I); 2: -2 + 4k + i - 4q - j
v 2: 4 - 3N - 2k
p 2: Ij - il
(A.9)
(A.I0)
(A.11)
Applying eq. (A.4) and using the (obvious) inequality j - i 2: -Ij - il one can justify
eq. (A.I0) also for that case.
3. q 2: k + 1
One can easily prove eq. (A.I0) far this case using eq. (A.5) and noting, that j -i 2: -3.
• E()..4k +i ) .. 4k+i Az'fl
v 2: 4 - 3N - 2k + l 2: 4 - 3N - 2k . (A.12)
This follows immediately from eq. (3.214). Here, the first inequality becomes the exact
equality when only the pion kinetic energy is taken into account in E()..4k +i ).
• )..4k +iAZ'flHK'fl
v = 4 - 3N - 2k + l + I); • (A.13)
There is no non-vanishing operator 'flHK'fl with I); < O. That is why the inequality (A.I0) is
again valid.