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

plug eq. (4.72) into eq. (4.70) and obtain
4.4 J , T -coupling of the m-scheme matrix elements
〈[(ja, jb)Jab, jc]J M, [(ta, tb)tab, tc]TMT |V NNN |[(j ′ a , j′ b )J ′ ab , j′ c ]J M, [(ta, tb)t ′ ab , tc]TMT 〉
=
ma,mb
Mab,mc
×
×
mta,mt b
mt ab ,mtc
ja jb
ma mb
j ′ a j′ b
m ′ a m ′ b
Jab
Mab
J ′ ab
M ′ ab
m ′ a ,m′ b
M ′ ab ,m′ c
m ′ ta ,m′ t b
m ′ t ab ,m ′ tc
Jab jc
Mab mc
J ′ ab j′ c
M ′ ab m′ c
J
M
J
M
ta ta
mta mtb
ta ta
m ′ ta m′ tb
tab
mtab
t
′ ab
m ′
tab
tab tc
mtab mtc
t ′ ab tc
m ′ tab m′ tc
× a〈abc|VNNN|a ′ b ′ c ′ 〉a . (4.73)
If we now look back at a〈abc|VNNN|a ′ b ′ c ′ 〉a in eq. (4.69), we recognize that fur-
ther simplifications are possible. Therefore, we move the unprimed terms inclu-
sive the corresponding sums of eq. (4.73) inside the unprimed ˜ T -coefficient of the
m-scheme matrix element and the primed ones in the primed ˜ T -coefficient. We
present the simplifications by showing only the relevant part of the m-scheme ma-
trix element and ˜ T -coefficient, namely the sum over Jab, J , tab, T and the Clebsch-
Gordan coefficients. Furthermore, we restrict ourselves to the simplification of the
unprimed part from above, since the primed ones can be simplified by analogous
steps.
The relevant part of the unprimed ˜ T and the first four Clebsch-Gordans from
eq. (4.73) with the corresponding sums read
ma,mb
Mab,mc
mta,mt b
ja jb Jab
×
ma mb
Mab
×
¯
J
¯Jab
¯tab
¯T
mt ab ,mtc
Jab jc
Mab mc
ja jb
ma mb
J
M
¯Jab
Mab
ta ta
mta mtb
Jab jc
Mab mc
tab
mtab
J¯
M
tab tc
mtab mtc
ta ta
mta mtb
¯tab
mtab
T
MT
T
MT
T
MT
tab tc
mtab mtc
¯T
MT
.
(4.74)
Here, J¯ is the sum over J from eq. (4.69) and ¯tab , T¯ are also given in eq. (4.69),
since they are implicit in
α . The summation Jab
¯ is the one from inside the
˜T -coefficient. Moreover, the projection quantum numbers do not require a tilde,
since they are constrained to be the sum of the single-particle projection quantum
J. Langhammer 63