The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.6. Application to chiral invariant Hamiltonians
number
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4 5 6
order r
Figure 3.5: Recursive prescription for calculating the operator ,\4k+i Ar]. The dashed
line shows a sequence of recursive steps. The large circles correspond to states with
4k pions (i = 0).
Thus, we again count the power of Q for the terms entering eq. (3.206) starting from its minimal
value. In appendix B we check what kind of operators ,\4k+i AIr] contribute to eq. (3.206) at each
fixed order r. That equation can be expressed as
85
(3.217)
Here, E('\ 4k+i) denotes the free energy of particles in the state ,\ 4k+i. In the curly brackets we
have written in symbolic form all remaining terms of eq. (3.206). For i = 1,2 they contain only
operators of the type ,\4k +i AIr] with
4k + i < 4k + i for l = r , or l < r (3.218)
As an example, we regard ,\4k+iH,\4 k +iAlr] with i = 1, 2. For 4k + i < 4k
+ i it follows from