4.4 Derivation of Three-Body Scattering Amplitude …
53
[
−1
c − h c ( p)] = m
a
d
p
2
− 2md E −
d
a
α
2
23
2π
2
(α 23 +
ap 2 /d − 2m E)
(4.36)
Since E =
k
2
2μ 1
−
α
2
23
2μ 23
=
k
2
2md
−
α
2
23
2ma
, the factor ( p
2
− 2md E −
d
a
α
2
23 ) simplifies to
( p
2
− k
2
), the inverse of which plays the role of a propagator for an outgoing wave
of the neutron relative to the n-core bound state.
The coefficient, [
−1
n − h n ( p)], of F(p) in Eq. (4.33) can, in a similar fashion, be
also simplified as
[
−1
n − h n ( p)] =
mλ
−1
n − m
d q
g
2
(q)
q 2 + p 2 /2a − m E
In the limit g( p) → 1, the above expression simplifies to
[
−1
n − h n ( p)] g( p)→1 → 2π
2 m
−
1
a nn
+
p 2
2a
− m E
(4.37)
We now substitute these simplifying factors in the expressions
[
−1
c − h c ( p)] = m
a
d
p
2
− 2md E −
d
a
α
2
23
2π
2
(α 23 +
ap 2 /d − 2ma E)
(4.38)
[
−1
n − h n ( p)] = m(2π
2
)
−
1
a nn
+
p 2 /2a − m E
(4.39)
to write in the spectator functions G(p) and F(p),
m
a
d
( p
2
− k
2
)
2π
2
(α 23 +
ap 2 /d − 2ma E)
G(
p)
≡
d q K 3 (
p,
q; E)G( q) +
d q K 2 ( p, q; E)F( q)
(4.40)
and
m(2π
2
)
−
1
a nn
+
p 2 /2a − m E
F(
p) ≡ 2
d q K 1 (
p,
q; E)G( q)
(4.41)
We now apply the boundary conditions
G(
p) = (2π)
3
δ(
p −
k) +
4π f k (
p)
( p 2 − k 2 − iε)
F(
p) =
4π b k (
p)
( p 2 − k 2 − iε)