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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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Appendix B<br />

Operators contribut<strong>in</strong>g to eq. (3.206)<br />

at order r<br />

In this appendix, we work out <strong>in</strong> detail which operators ).a Ar" actually appear <strong>in</strong> eq. (3.206) at<br />

order r, which is def<strong>in</strong>ed <strong>in</strong> eq. (3.216). For that we will exam<strong>in</strong>e below all terms that enter this<br />

equation. We f<strong>in</strong>d<br />

• ).4k+iHK,).4 q + j A/1]<br />

One obta<strong>in</strong>s from (A.8) and (3.216) the follow<strong>in</strong>g identity:<br />

Let us first consider the case i # 0:<br />

1. q :::; k-1<br />

Us<strong>in</strong>g eqs. (A.9), (B.1) one gets<br />

2. q = k<br />

l = r + 2(k - q) + i - j - K,<br />

l :::; r - 2k + 2q + 2 :::; r .<br />

(a) j < i<br />

It follows from eqs. (A.ll), (B.1), that<br />

(b) J = Z<br />

In this case eq. (B.1) takes the form<br />

l :::; r - Ij - i 1 + i - j :::; r .<br />

due to the <strong>in</strong>equalities (A.3), (A.4) and (A.6).<br />

(c) j 2:: i + 1<br />

<strong>The</strong> first <strong>in</strong>equality of eq. (B.3) leads to<br />

3. q = k + 1<br />

l :::; r - 2j + 2i :::; r - 2 .<br />

175<br />

(B.1)<br />

(B.2)<br />

(B.3)<br />

(B.4)<br />

(B.5)

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