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

4 Three-body Jacobi matrix element transformation into the m-scheme
ing. Finally, in subsection 4.4, we present the solution of this problem by slightly
changing our strategy into the calculation of matrix elements that maintain a total
coupled angular momentum and isospin.
4.1 Matrix elements of the three-nucleon interaction at N2LO in
the m-scheme
In the following, we derive the formula for the transformation of three-particle
interaction matrix elements from the antisymmetrized Jacobi states |EJTi〉 into
the antisymmetrized m-scheme basis
〈EJTi|V NNN |E ′ JTi ′ 〉 −→ a〈abc|V NNN |a ′ b ′ c ′ 〉a . (4.2)
We start with a non-antisymmetrized product state of (ls)-coupled single-particle
harmonic oscillator states
|abc〉 = |(nala, sa)jama, (nblb, sb)jbmb, (nclc, sc)jcmc, tamtatbmtb tcmtc〉 . (4.3)
In the first step, we couple the single-particle angular momenta ji to total angular
momentum J of the three nucleons
|abc〉 =
ja jb Jab Jab jc J
M
Jab,J
ma mb
Mab
Mab mc
× |{[(nala, sa)ja, (nblb, sb)jb]Jab, (nclc, sc)jc}J M, tamtatbmtb tcmtc〉 ,
(4.4)
with Mab = ma + mb and M = Mab + mc. We used the Clebsch-Gordan coefficients
defined in eq. (3.5) and eliminated the sums over projection quantum numbers
using eq. (3.9).
In the next step we make use of the identity
1 =
ncm,lcm
α
J ′ ,M ′
{|ncmlcm〉 ⊗ |α〉} J ′ M ′
{〈ncmlcm| ⊗ 〈α|} J ′ M ′
where ncm, lcm denote center-of-mass quantum numbers and
, (4.5)
|α〉 = |[(n12l12, sab)j12, (n3l3, sc)j3]JMJ, (tabtc)TMT 〉 (4.6)
again denotes the Jacobi state as already introduced in section 3.4. Here, one has
to be careful, since |α〉 does not contain the MJ quantum number if its total an-
gular momentum is coupled, as in eq. (4.5). There, the curly brackets indicate the
coupling of the center-of-mass orbital angular momentum with the total angular
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