K
Example of a Cross Section Calculation
+
In this appendix we outline in more detail the calculation of the e
− s elastic
scattering cross section in section 8.3.2. The standard factors for the unpolarized cross section lead to the expression
d¯ σ =
1
1 ∑
|M e − s + (s, s
′ )|
2 dLips(s; k
′ , p
′ )
(K.1)
4Eω|v| 2 '
ss
=
1
4[(k.p) 2 − m 2 M 2 ] 1/2
1
2
∑
ss '
|M e − s + (s, s
′ )|
2 dLips(s; k
′ , p
′ ) (K.2)
using the result of problem 6.9, and the definition of Lorentz-invariant phase
space:
′
d
3
p
d
3
k
′
′
dLips(s; k
′ , p
′ ) ≡ (2π)
4 δ
4 (k
′ + p − k − p)
.
(K.3)
(2π) 3 2E ′ (2π) 3 2ω ′
Instead of evaluating the matrix element and phase space integral in the CM
frame, or writing the result in invariant form, we shall perform the calculation
entirely in the ‘laboratory’ frame, defined as the frame in which the target (i.e.
the s-particle) is at rest:
p
μ = (M, 0)
(K.4)
where M is the s-particle mass. Let us look in some detail at the ‘laboratory’
frame kinematics for elastic scattering (figure K.1). Conservation of energy
and momentum in the form
′ 2
p = (p + q)
2
(K.5)
′
allows us to eliminate p to obtain the elastic scattering condition
2
2p · q + q = 0
(K.6)
or
2p · q = Q
2
(K.7)
if we introduce the positive quantity
2
Q
2 = −q
(K.8)
for a scattering process.
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