54
4 Effective Field Theory
to the Eqs. (4.40) and (4.41) and setting the limit g( p)/ f ( p) → 1, we get the
equations for the scattering amplitude for n-dimer scattering
a
d
(2π 2 )
[α 23 +
ap 2 /d − 2ma E]
(4π) f k (
p) = (2π) 3 K 1 (
p,
k; E)
+ (4π)
d q
K 1 (
p,
q; E) f k ( q)
(q 2 − k 2 − iε)
+ (4π)
d q
K 2 (
p,
q; E)b( q, E)
(q 2 − k 2 − iε)
or
a
d
1
[α 23 +
ap 2 /d − 2ma E]
f k ( p) = K 1 (
p,
k; E) +
1
2π 2
d q
K 1 (
p,
q; E) f k ( q)
(q 2 − k 2 − iε)
+
1
(2π 2 )
d q
K 2 (
p,
q; E)b k ( q)
(q 2 − k 2 − iε)
(4.42)
and
(−1/a nn +
p 2 /2a − m E)b k (
p)
( p 2 − k 2 )
= 2K 3 (
p,
k ; E) + 2(
1
2π 2 )
d q
K 3 (
p,
q; E) f k ( q)
(q 2 − k 2 − iε)
(4.43)
After performing the angular integration, the expressions (4.42) and (4.43) for the
s-wave scattering amplitude get reduced to
a
d
1
[α 23 +
ap 2 /d − 2ma E]
f 0 ( p, k; E) = Z 1 ( p, k; E)
+
1
π
q 2 dq Z 1 (q, k; E)
f 0 (q, k; E)
(q 2 − k 2 − iε)
;
+
1
π
q 2 dq Z 2 (q, k; E)
b 0 (q, k; E)
(q 2 − k 2 − iε)
(4.44)
and
(−1/a nn +
p 2 /2a − m E)b 0 ( p, k; E)
( p 2 − k 2 )
= 2Z 3 ( p, k; E)
+ 2
1
π
q
2 dq
Z 3 ( p, q; E) f 0 (q, k; E)
(q 2 − k 2 − iε)
(4.45)
where