176
6. Quantum Field Theory II: Interacting Scalar Fields
′
The energy E is given by
A
′
2
′2
E A = (m A + p A )
1/2
(6.117)
so that
′
′
′
′
E A dE = |p A | d|p A |.
(6.118)
A
With all these changes we arrive at the result (valid in any frame)
′
′
1 |p |dE
′
′
A
A
′
′
dLips(s; p A , p B ) =
dΩ δ(E A + E B − E A − E B ).
(6.119)
′
(4π) 2 E B
′
′
We now specialize to the CM frame for which p A = p = −p B , p = p =
A
′
−p B , and
′
2
′2 )
1/2
′
2
′2 )
1/2
E A = (m A + p
E B = (m B + p
(6.120)
so that
′
′
′
′
′
|
E A dE = |p
′
| d|p = E B dE B .
(6.121)
A
′
′
Introduce the variable W
′ = E A + E (note that W
′ is only constrained
B
to equal the total energy W = E A + E B after the integral over the energyconserving δ-function has been performed). Then (as in (6.62))
W
′
|p
′
| d|p
′
|
W
′
′
′
′
dW
′ = dE A + dE =
=
dE
(6.122)
B
′
′
′
A
E E
E
A B
B
where we have used (6.121) in each of the last two steps. Thus the factor
′
dE
′
A
′
′
|p A |
δ(E A + E B − E A − E B )
(6.123)
′
E B
becomes
dW
′
|p
′
|
δ(W − W
′ )
(6.124)
W ′
which reduces to
|p|/W
after integrating over W
′ , since the energy-conservation relation forces |p
′
| =
|p|. We arrive at the important result
1 |p|
′
′
dLips(s; p A , p B ) =
dΩ
(6.125)
(4π) 2 W
for the two-body phase space in the CM frame.
The last piece in the puzzle is the evaluation of the flux factor (6.112) in
the CM frame. In the CM we have
p A · p B = (E A , p) · (E B , −p)
(6.126)
2
= E A E B + p
(6.127)