48
4 Effective Field Theory
g(p) is taken as 1 in the limit so as to reduce the integral equation for the bosondimer scattering amplitude exactly that obtained in the effective field theory with
short-range interaction alone. The main thrust of this approach is to demonstrate,
through a detailed analysis studying n-d scattering, that the presence of eigenvalues
in the attractive channel in three-body scattering using effective field theory can
be removed by one subtraction from the kernel to obtain a renormalization group
invariant scattering amplitude.
As we are basically interested in investigating the structural properties of halo
nuclei within the framework of three-body system using separable potential, the
above approach by Afnan and Phillips can be easily tuned particularly for studying
the resonant structure of n–
19 C(n–dimer(n–
18 C)) scattering at low energies, using
effective field theory.
4.4 Derivation of Three-Body Scattering Amplitude
in Effective Field Theory Approaching from Separable
Potentials
To derive the equation for the scattering amplitude in effective field theory from
s-state separable potential in the three-boson system, we start from Eq. (2.74) for
the spectator function G(
p) of third particle in the presence of the other two bound
particles.
( 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)
,
(4.16)
where
H (3 p
2
/4 − m E) =
d q
g
2
(q)
(q 2 + α 2 )(q 2 + 3 p 2 /4 − m E)
(4.17)
Substituting (4.17) and using the boundary condition
G(
p) = (2π)
3
δ(
p −
k) +
4π f k (
p)
p 2 − k 2 − iε
(4.18)
we get the integral equation for the scattering amplitude, f k (
p),