The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

178 C. Small scale expansion within the projection formalism
• )..0 Al TJHt;, TJ
Equation (A.13) leads to the following inequality:
v 2 8 - 3N , (C.6)
where we have used the facts that min (l) = 2 for )..0 A1TJ and min (I\;) = 2 for TJHt;,TJ.
v 2 8 - 3N . (C.7)
v = 4 - 3N + 2q + j + h + l 2 + I\; 2 8 - 3N . (C.8)
To complete the modifications of the appendix A, we also need to consider ).. O -intermediate states.
Only two terms in eq. (3.206) may have such intermediate states:
• )..4k+iHt;,)"oA1TJ
We consider two different cases:
v = 4 - 3N - 4k - i + I\; + 1 . (C.9)
1. k = 0, i f. 0
Using the inequality (A.4) and the fact that 1 2 2 we obtain
2. k f. 0
In that case eq. (A.6) leads to
v 2 6 - 3N . (C.10)
v 2 4 - 3N . (C.ll)
v = 4 - 3N - 2k + h + l 2 + I\; 2:: 8 - 3N - 2k . (C.12)
We now switch to appendix B and consider which operators )..a A1TJ enter the decoupling equation
(3.206) at order r projected onto the state )..0•
• )..0 Ht;,)..4 q +j A1TJ
Again, we have to consider two cases:
1. q=O
One can use eq. (A.4) if j f. 0 and the fact that I\; 2 2 if j = 0 to obtain
l