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

4 Three-body Jacobi matrix element transformation into the m-scheme
part of the Hilbert space. Consequently, we now make use of the representation of
the antisymmetrizer that we worked out in section 3.4 in eq. (3.94)
A =
ncm,lcm
mcm
E,J,MJ T,MT
|ncmlcmmcm〉 ⊗ |EJMJTMTi〉〈ncmlcmmcm| ⊗ 〈EJMJTMTi| .
i
(4.11)
If we multiply eq. (4.10) with the antisymmetrizer from the left, we have to consider
the overlap
〈n ′ cml ′ cmm ′ cm|ncmlcmmcm〉〈EJ ′ M ′ JT ′ M ′ JT ′ M ′ Ti|α〉
= cα,i δJ ′ ,J δM ′ J ,MJ δT ′ ,T δM ′ T ,MT δn ′ cm,ncm δl ′ cm,lcm δm ′ cm,mcm δE,2n12+l12+2n3+l3 , (4.12)
with the coefficient of fractional parentage cα,i as defined in eq. (3.91). Using the
Kronecker deltas of eq. (4.12), we eliminate the corresponding summations and
obtain for the antisymmetrized m-scheme state
|abc〉a = √ 3!
Jab,J
α
lcm,ncm, MJ
mcm
ja jb Jab Jab jc J
×
ma mb
Mab Mab mc
M
⎛
⎞
a b c Jab J J
⎜
⎟
× T⎝
⎠
ncm lcm n12 l12 n3 l3
sab j12 j3 tab T MT
× cα,i |ncmlcmmcm〉 ⊗ |EJMJTMTi〉 .
i
lcm J
mcm MJ
J
M
(4.13)
The only remaining sum of (4.11) is the one with respect to i. The quantum num-
ber E is now constrained by E = 2n12 + l12 + 2n3 + l3. The additional factor √ 3!
shows up because of
|abc〉a = √ 3! A |abc〉 , (4.14)
and is needed to have a normalized state after the projection.
Now we are in the position to write down the interaction matrix element. We
use the hermitian adjoint of |abc〉a, as given in eq. (4.13), to sandwich the interac-
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