28
2 Scattering Theory of Three-Particle System
where
H (3 p
2
1 /4 − m E) =
d
q
g
2
(q)
(q 2 + α 2 ) (q 2 + 3 p
2
1 /4 − m E)
,
which clearly means that the function F(
p 1 ) exhibits a pole at p
2
1 = k
2 . Thus
redefining H (3 p
2
1 /4 − m E) F(
p 1 ) by G(
p 1 ) in Eq. (2.67), we get
( p
2
1 /2μ − k
2
/2μ ) G(
p 1 ) =
2
m
d
q
g( q + +
p 1 /2) g(
p 1 + +
q/2) G( q)
(q 2 + p
2
1 + +
q.
p 1 − m E) H (3q 2 /4 − m E)
(2.74)
It follows from this equation that to describe the scattering process, the function
G(
p 1 ) can be used to assign the proper boundary condition consisting of an incident
plane wave plus an outgoing scattered wave, which in momentum space means
G(
p 1 ) = (2π)
3
δ(
p 1 −
k) +
4π f k (
p 1 )
p
2
1 − k 2 − iε
(2.75)
Substituting (2.75) in (2.74) gives us the equation for the scattering amplitude
(3/4)4π f k (
p 1 ) = 2(2π)
3 N
2 g(
p 1 +
k/2) g(
k + +
p 1 /2)
( p
2
1 + k 2 + +
p 1 .
k − m E)
+ 2(4π)
d q
g( q + +
p 1 /2) g(
p 1 + +
q/2) f k ( q)
( p
2
1 + q 2 + +
p 1 . q − m E) (H (3q 2 /4 − m E)) (q 2 − k 2 − iε)
(2.76)
We have thus finally obtained a simple one variable integral equation for an
off-shell scattering amplitude to describe the scattering by bound state of identical
particles. The energy E is of course understood to have an imaginary term iE. As
regards the uniqueness problem for the boundary condition to obtain the amplitude
for the channel of scattering of particle ‘i’ by the bound ‘jk’ system, this can be
resolved in the Schrodinger equation approach purely on physical considerations.
For instance, considering the case of 3 nucleons taking into account the spin, i-spin
degrees of freedom, we consider a system consisting of (2n + 1p) corresponding to
the i-spin I = 1/2 and I z = 1/2, representing the scattering of neutron by deuteron.
If we have only s-state interaction between different pairs, we would then have
the interaction between singlet and triplet spin states. As a result, we would get
two coupled integral equations for the spectator functions F 1 (
p) and F 2 (
p). While
F 1 (
p) may represent the scattering of neutron by deuteron (
3 S 1 ) state, the function
F 2 (
p) will correspond to the scattering of neutron by a d
∗
(
1 S 0 ) state which is a
virtual (anti-bound) state. Thus, for n-d scattering only F 1 (
p) will be subject to the
2 Scattering Theory of Three-Particle System
where
H (3 p
2
1 /4 − m E) =
d
q
g
2
(q)
(q 2 + α 2 ) (q 2 + 3 p
2
1 /4 − m E)
,
which clearly means that the function F(
p 1 ) exhibits a pole at p
2
1 = k
2 . Thus
redefining H (3 p
2
1 /4 − m E) F(
p 1 ) by G(
p 1 ) in Eq. (2.67), we get
( p
2
1 /2μ − k
2
/2μ ) G(
p 1 ) =
2
m
d
q
g( q + +
p 1 /2) g(
p 1 + +
q/2) G( q)
(q 2 + p
2
1 + +
q.
p 1 − m E) H (3q 2 /4 − m E)
(2.74)
It follows from this equation that to describe the scattering process, the function
G(
p 1 ) can be used to assign the proper boundary condition consisting of an incident
plane wave plus an outgoing scattered wave, which in momentum space means
G(
p 1 ) = (2π)
3
δ(
p 1 −
k) +
4π f k (
p 1 )
p
2
1 − k 2 − iε
(2.75)
Substituting (2.75) in (2.74) gives us the equation for the scattering amplitude
(3/4)4π f k (
p 1 ) = 2(2π)
3 N
2 g(
p 1 +
k/2) g(
k + +
p 1 /2)
( p
2
1 + k 2 + +
p 1 .
k − m E)
+ 2(4π)
d q
g( q + +
p 1 /2) g(
p 1 + +
q/2) f k ( q)
( p
2
1 + q 2 + +
p 1 . q − m E) (H (3q 2 /4 − m E)) (q 2 − k 2 − iε)
(2.76)
We have thus finally obtained a simple one variable integral equation for an
off-shell scattering amplitude to describe the scattering by bound state of identical
particles. The energy E is of course understood to have an imaginary term iE. As
regards the uniqueness problem for the boundary condition to obtain the amplitude
for the channel of scattering of particle ‘i’ by the bound ‘jk’ system, this can be
resolved in the Schrodinger equation approach purely on physical considerations.
For instance, considering the case of 3 nucleons taking into account the spin, i-spin
degrees of freedom, we consider a system consisting of (2n + 1p) corresponding to
the i-spin I = 1/2 and I z = 1/2, representing the scattering of neutron by deuteron.
If we have only s-state interaction between different pairs, we would then have
the interaction between singlet and triplet spin states. As a result, we would get
two coupled integral equations for the spectator functions F 1 (
p) and F 2 (
p). While
F 1 (
p) may represent the scattering of neutron by deuteron (
3 S 1 ) state, the function
F 2 (
p) will correspond to the scattering of neutron by a d
∗
(
1 S 0 ) state which is a
virtual (anti-bound) state. Thus, for n-d scattering only F 1 (
p) will be subject to the