8.17
Problems
265
(a) Derive an expression for the spin-averaged differential cross section
for lowest-order e
− μ
− scattering in the laboratory frame, defined
by p
μ = (M, 0) where M is now the muon mass, and show that it
may be written in the form
(
)
dσ
dσ
=
[1 − (q
2 /2M
2 ) tan
2 (θ/2)]
dΩ
dΩ ns
+
where the ‘no-structure’ cross section is that of e
− s scattering
(appendix K) and the electron mass has been neglected.
(b) Neglecting all masses, evaluate the spin-averaged expression (8.184)
in terms of s, t and u and use the result
∑
dσ
1 1
′
=
|M e − μ − (r, s; r , s
′ )|
2
dt
16πs 2 4
r,r ' ;s,s '
to show that the e
− μ
− cross section may be written in the form
(
)
dσ
4πα
2 1
u
2
=
1 +
.
dt
t 2 2
s 2
Show also that by introducing the variable y, defined in terms of
laboratory variables by y = (k − k
′ )/k, this reduces to the result
dσ
4πα
2 1
=
s [1 + (1 − y)
2 ].
dy
t 2 2
8.18 Consider the process e
+ e
−
→ μ
+ μ
− in the CM frame.
(a) Draw the lowest-order Feynman diagram and write down the corresponding amplitude.
(b) Show that the spin-averaged squared matrix element has the form
(4πα)
2
|M| 2 =
L(e) μν L(μ)
μν
q 4
where q
2 is the square of the total CM energy, and L(e) depends on
the e
− and e
+ momenta and L(μ) on those of the μ
+ , μ
− .
(c) Evaluate the traces and the tensor contraction (neglecting lepton
masses): (i) directly, using the trace theorems; and (ii) by using
crossing symmetry and the results of section 8.7 for e
− μ
− scattering.
Hence show that
|M| 2 = (4πα)
2 (1 + cos
2 θ)