415
K.2. Evaluation of two-body Lorentz-invariant phase space
and the δ-function requires that k
′ is determined from k and θ by the elastic
scattering condition
k
′ =
k
≡ k
′ (cos θ).
(K.36)
1 + (2k/M ) sin
2 (θ/2)
The integral (K.32) becomes
∫
k
′
1
dΩ dk
′
1
δ[k
′
− k
′ (cos θ)]
(K.37)
E ′
(4π) 2
|df /dk ′ | k ' =k ' (cos θ)
and, after some juggling, df /dk
′ evaluated at k
′ = k
′ (cos θ) may be written
as
df
M k
=
.
(K.38)
dk ′
|
|
|
|
E ′ k ′
k ' =k ' (cos θ)
Thus we obtain finally the result
k
′ 2
dLips(s; k
′ , p
′ ) =
1
dΩ
(K.39)
(4π) 2 M k
for two-body elastic scattering in terms of ‘laboratory’ variables, neglecting
lepton masses.
Putting all these elements together yields the advertised result
(
)
dσ
d¯ σ
α
2
k
′
≡
=
cos
2 (θ/2).
(K.40)
dΩ
dΩ
4k 2 sin
4 (θ/2) k
ns
As a final twist to this calculation let us consider the change of variables from
dΩ to dq
2 in this elastic scattering example. In the unpolarized case
dΩ = 2πd(cos θ)
(K.41)
and
2
q = −2kk
′ (1 − cos θ)
(K.42)
where
k
k
′ =
.
(K.43)
1 + (2k/M ) sin
2 (θ/2)
Thus, since k
′ and cos θ are not independent variables, we have
dk
′
dq
2 = 2kk
′ d(cos θ) + (1 − cos θ)(−2k)
d(cos θ).
(K.44)
d(cos θ)
From (K.20) we find
dk
′
k
′ 2
=
(K.45)
d(cos θ)
M