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

3 Mathematical basics
Now we make use of the fact that the same transformation matrix combines the
coordinates in the following way
d
⎛
−r
R
1+d
= ⎝
1
1+d −
Note that R =
1
1+d
d
1+d
⎞
⎠
r2
−r1
. (3.49)
d
1+dr1
1 + 1+dr2 and r =
1
1+dr1
d − 1+dr2 still are given as in
eq. (3.43). This must always be fulfilled for all symmetry relations we derive in the
following. Furthermore, we know the following relations
〈(n1l1, n2l2)Λλ|r1,r2〉 = (−1) l1 (−1) l1+l2−Λ 〈(n2l2, n1l1)Λλ|r2, −r1〉 , (3.50)
〈(EL, el)Λλ| R,r〉 = (−1) l (−1) l+L−Λ 〈(el, EL)Λλ| − r, R〉 . (3.51)
The first phase factor with one orbital angular momentum in the exponent cor-
responds to the behavior of the harmonic oscillator wave functions under parity
transformation. The second one, with three orbital angular momenta in the expo-
nent, is exactly the one we know from eq. (3.12) and shows up because we reversed
the coupling order of the angular momenta. Using these relations we can rewrite
eq. (3.48) as
〈〈n1l1, n2l2; Λ|NL, nl〉〉d
1
¨
=
2Λ + 1
λ
1
¨
=
2Λ + 1
λ
d 3 r1d 3 r2〈(n2l2, n1l1)Λλ|r2, −r1〉〈−r, R|(nl, NL)Λλ〉(−1) l2+L
d 3 r1d 3 r2〈(n2l2, n1l1)Λλ|r2, −r1〉〈r2, −r1|(nl, NL)Λλ〉(−1) l2+L
= (−1) l2+L 〈〈n2l2, n1l1; Λ|nl, NL〉〉d , (3.52)
where we used the identity
¨
1 =
d 3 r1d 3 r2|r2, −r1〉〈r2, −r1| . (3.53)
To simplify the phase factor we used the fact that orbital angular momenta are
integer and so (−1) 2l ≡ 1.
For the next symmetry relation we use that the transformation
⎛
R
= ⎝
−r
1
1+d
d
1+d −
d
1+d
1
1+d
⎞
r2 ⎠
r1
(3.54)
connects the coordinates in the correct way, where we also changed d → 1 in the d
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