Consistent chiral three-nucleon interactions in ... - Theory Center

3 Mathematical basics
We can then rewrite eq. (3.66) into
〈〈n1l1, n2l2; Λ|nl, NL〉〉 alt
d
1
¨
=
2Λ + 1
λ
1
¨
=
2Λ + 1
d 3 r1d 3 r2〈(n2l2, n1l1)Λλ|r2,r1〉〈−r, R|(nl, NL)Λλ〉(−1) l1+l2−Λ (−1) l
d 3 r1d 3 r2〈(n2l2, n1l1)Λλ|r2,r1〉〈r2,r1|(nl, NL)Λλ〉(−1) l1+l2−Λ (−1) l
λ
= (−1) 2l+L−Λ 〈〈n2l2, n1l1; Λ|nl, NL〉〉 alt
1
d
= (−1) L−Λ 〈〈n2l2, n1l1; Λ|nl, NL〉〉 alt
1
d
Finally, for the last symmetry relation we use the transformation matrix
d
⎛
r
R
1+d
= ⎝
1
1+d −
1
1+d
d
1+d
(3.74)
⎞
⎠
r1
−r2
, (3.75)
where also d → 1 has been used. The relations we need here are given by
d
〈(nl, NL)Λλ|r, R〉 = (−1) L+l−Λ 〈(NL, nl)Λλ| R,r〉 (3.76)
〈n2l2|r2〉 = (−1) l2 〈n2l2| − r2〉 , (3.77)
and we rewrite eq. (3.66) as
〈〈n1l1, e2l2; Λ|nl, NL〉〉 alt
d
1
¨
=
2Λ + 1
d 3 r1d 3 r2〈(n1l1, n2l2)Λλ|r1, −r2〉〈 R,r|(NL, nl)Λλ〉(−1) L+l−Λ+l2
λ
= (−1) 2l2+l1−Λ
〈〈n1l1, n2l2; Λ|NL, nl〉〉 alt
1
d
= (−1) l1−Λ
〈〈n1l1, n2l2; Λ|NL, nl〉〉 alt
1
d
. (3.78)
So we derived the analogous relations as in section 3.3.2. The alternative HOBs are
real as well, so we have the following summarized relations
〈〈n1l1, n2l2; Λ|nl, NL〉〉 alt
d
= 〈〈nl, NL; Λ|n1l1, n2l2〉〉 alt
d
= (−1) l1+l 〈〈n2l2, n1l1; Λ|NL, nl〉〉 alt
d
= (−1) L−Λ 〈〈n2l2, n1l1; Λ|nl, NL〉〉 alt
1
d
= (−1) l1−Λ
〈〈n1l1, n2l2; Λ|NL, nl〉〉 alt
1 .
d
These and more symmetry relations can be found in [22].
28
(3.79)