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8.7. Electron muon elastic scattering
FIGURE 8.17
One-photon exchange amplitude in e
− μ
− scattering.
We shall consider e
− μ
− elastic scattering: our notation is indicated in figure 8.16. In the lowest order of perturbation theory – the one-photon exchange
approximation – we can draw the relevant Feynman graph for this process.
This is shown in figure 8.17. All the elements for the graph have been met
before and so we can immediately write down the invariant amplitude which
now depends on four spin labels:
′
′
M e − μ − (r, s; r , s
′ ) = eu ¯(k
′ , s
′ )γ μ u(k, s)(g
μν /q
2 )eu ¯(p , r
′ )γ ν u(p, r). (8.182)
Although experiments with polarized leptons are not uncommon, we shall
only be concerned with the unpolarized cross section
∑
′
d¯ σ ∼
1
|M e − μ − (r, s; r , s
′ )|
2 .
(8.183)
4
r,r ' ;s,s '
+
We perform the same manipulations as in our e
− s example and the cross
section reduces to a factorized form involving two traces:
( ) 2 (
)
∑
2
1
e
1
′
′
|M e − μ − (r, s; r , s
′ )|
2
=
Tr[(k / + m)γ μ (k / + m)γ ν ]
4
q 2
2
r,r ' ;s,s '
′
× {
1 Tr[(p / + M )γ
μ (p / + M )γ
ν ]} (8.184)
2
= (e
2 /q
2 )
2 L μν M
μν
(8.185)
where L μν is the ‘electron tensor’ calculated before (see (8.119)):
L μν = 2[k μ
′ k ν + k ν
′ k μ + (q
2 /2)g μν ]
(8.186)
but now M
μν is the appropriate tensor for the muon coupling, with the same
structure as L μν :
′ μ ν
′ ν
μν ].
M
μν = 2[p p + p p
μ + (q
2 /2)g
(8.187)