The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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Appendix G<br />
Partial wave decomposition of the<br />
N N potential<br />
In this appendix we would like to deseribe the partial wave deeomposition of the two-nucleon<br />
potential. For that we first rewrite the potential V <strong>in</strong> the form<br />
V =<br />
Vc<br />
+ Va- ih · ih + VSL i �(o\ + 5 2 ) · Cf x ifJ + Va-L 51 · (ifx k) 5 2 · (if x k)<br />
+ Va-q (51 · ifJ (5 2 · ifJ + Va-k (51 . k) (5 2 · k) (G.l)<br />
with six functions Vc (p,p',z), ... , Va-k (P,P',z) depend<strong>in</strong>g on p == Ipl, p' == Ip'l and the eos<strong>in</strong>e<br />
of the angle between the two momenta is ealled z. <strong>The</strong>se functions may depend on the isosp<strong>in</strong><br />
matriees T as weIl. To perform the partial wave deeomposition of V, i. e. to express it <strong>in</strong> the<br />
standard lsj representation, we have followed the steps of ref. [193]. In particular, we start from<br />
the helicity state representation Iß A1A<br />
2 ), where ß = vip and Al and A<br />
2 are the helicity quantum<br />
numbers eorrespond<strong>in</strong>g to nucleons 1 and 2, respeetively. We then expressed the potential <strong>in</strong><br />
the IjmA1A<br />
2 ) representation us<strong>in</strong>g the transformation matrix (ß A1A2IjmA1A2 ), given <strong>in</strong> ref. [193].<br />
<strong>The</strong> f<strong>in</strong>al step is to switeh to the Ilsj) representation. <strong>The</strong> eorrespond<strong>in</strong>g transformation matrix<br />
(lsjmlJmA1A<br />
2 ) is given <strong>in</strong> refs. [194], [193].<br />
For j > 0, we obta<strong>in</strong>ed the follow<strong>in</strong>g expressions for the non-vanish<strong>in</strong>g matrix elements <strong>in</strong> the<br />
Ilsj) representation:<br />
(jOj I V IjOj)<br />
(jljlVljlj)<br />
27r [ 1<br />
1 dz {Vc - 3Va- + p,2 p2(z2 - 1)Va-L - q2Va-q - k2Va- k} Pj (z) ,<br />
27r [ 1<br />
1 dz {[Vc + Va- + 2P'PZVSL - p,2 p2(1 + 3z2)Va-L + 4k2Va-q + lq2Va-k]<br />
X Pj (z) + [-p'p VSL + 2p, 2 p2zVa-L - 2p'p (Va-q - l Va- k)]<br />
x (Pj-1(z) + Pj +1(Z)) } ,<br />
(j ± 1, IjlVlj ± 1, Ij) 27r [ 1<br />
1 dz {p'p [ -VSL ± 2j � 1 ( -p'PZVa-L + Va-q - l Va-k)]<br />
x Pj(z) + [Vc + Va- + p'pzVsL + p, 2 p2(1 _ z2)Va-L (G.2)<br />
192