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

3 Mathematical basics
up the (LS)-coupling using a Clebsch-Gordan coefficient. We arrive at
|[(n12l12, sab)j12, (n3l3, sc)j3]JMJ, (tabtc)TMT 〉
=
ˆj12
L,ML S,MS
ˆj3 ˆ L ˆ ⎧ ⎫
⎪⎨ l12 sab j12⎪⎬
L S
S l3 sc j3
⎪⎩ ⎪⎭ ML MS
L S J
J
MJ
× |(n12l12, n3l3)LML, (sab, sc)SMS, (tabtc)TMT 〉 .
The matrix element of T23 then reads
〈[(n ′ 12 l′ 12 , s′ ab )j′ 12 , (n′ 3 l′ 3 , sc)j ′ 3 ]J ′ M ′ J , (t′ ab tc)T ′ M ′ T |
=
L ′ ,S ′ M ′ L ,M′ L,S ML,MS
S
⎧ ⎫⎧
⎪⎨ l12 sab j12⎪⎬
⎪⎨
× l3 sc j3
⎪⎩ ⎪⎭ ⎪⎩
L S J
ˆj ′ 12 ˆj ′ 3 ˆ L ′ ˆ S ′ˆj12 ˆj3 ˆ L ˆ S
l ′ 12 s ′ ab j′ 12
l ′ 3 sc j ′ 3
L ′ S ′ J ′
(3.102)
× T23|[(n12l12, sab)j12, (n3l3, sc)j3]JMJ, (tabtc)TMT 〉
⎫
⎪⎬
L S
⎪⎭ ML MS
J
MJ
× 〈(n ′ 12 l′ 12 , n′ 3 l′ 3 )L′ M ′ L |T23|(n12l12, n3l3)LML〉
L ′ S ′
M ′ L M ′ S
J
′
× 〈(s ′ ab , sc)S ′ M ′ S |T23|(sab, sc)SMS〉 〈(t ′ ab , tc)T ′ M ′ T |T23|(tab, tc)TMT 〉 . (3.103)
We note that the structure of the spin and isospin matrix element is identical, so it
is sufficient to work out one of them. We investigate the isospin matrix element
〈(t ′ ab , tc)T ′ M ′ T |T23|(tab, tc)TMT 〉 , (3.104)
by considering the action of the transposition operator on the ket
T23|(tab, tc)TMT 〉
=
mtamt b mtc
=
mtamt b mtc
ta tb
mta mtb
ta tb
mta mtb
tab
mtab
tab
mtab
tab tc
mtab mtc
tab tc
mtab mtc
T
MT
T
MT
M ′ J
T23|tamta, tbmtb , tcmtc〉 (3.105)
|tamta, tcmtc, tbmtb 〉 . (3.106)
Now particle 3 is in the state |tbmtb 〉 and couples its isospin with particle 1 to the
same quantum number tab as before. This is the same situation as if we rewrite the
formula with exchanged quantum numbers tb ↔ tc
T23|[(ta, tb)tab, tc]TMT 〉 = |[(ta, tc)tab, tb]TMT 〉 . (3.107)
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