50
4 Effective Field Theory
[λ
−1
− h(3 p
2
/4 − m E)]F(
p) = 2
d q
g( q + +
p/2)g( ¯
p + +
q/2)
q 2 + p 2 + +
q.
p − m E − iε
F( q) (4.23)
where
h(3 p
2
/4 − m E) =
d q
g
2
(q)
(q 2 + 3 p 2 /4 − m E)
(4.24)
and applying the boundary condition
F(
p) = (2π)
3
δ(
p −
k) +
4πa k (
p,
k; E)
p 2 − k 2 − iε
(4.25)
we get the equation for the scattering amplitude
(3/4)4π H
3
4
p
2
− m E
a k (
p,
k; E) = 2(2π)
3 g(
p +
k/2)g(
k + +
p/2)
( p 2 + k 2 + +
p.
k − m E)
+ 2(4π)
d q
g( q + +
p/2)g(
p + +
q/2)a k ( q,
k; E)
( p 2 + q 2 + +
p. q − m E)(q 2 − k 2 − iε)
(4.26)
Again, to get the equation for the scattering amplitude for short-range interaction
in EFT, we take the limit g( p) → 1, and have
(3/4)4π
2π 2
(α +
3 p 2 /4 − m E)
a(
p,
k; E) = 2(2π) 3
1
( p 2 + k 2 + +
p ·
k − m E)
+ 2(4π)
d q
a( q,
k; E)
( p 2 + q 2 + +
p · ·
q − m E)(q 2 − k 2 − iε)
(4.27)
and get the s-wave scattering amplitude
3
8(α +
3 p 2 /4 − m E)
a 0 ( p, k; E) = M( p, k; E)
+
2
π
q 2 dq
M(q, k; E)a 0 (q, k; E)
(q 2 − k 2 − iε)
,
(4.28)
where
M(q, k; E) =
1
2qk
ln
q
2
+ k
2
+ qk − m E
q 2 + k 2 − qk − m E
(4.29)
Equation (4.28) is the familiar Skorniakov-Ter Marterosan equation [36] for the
three-boson system describing s-wave boson-dimer scattering amplitude.