The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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