256
8. Elementary Processes in Scalar and Spinor Electrodynamics
To evaluate the cross section we must perform the ‘contraction’ L μν M
μν .
A useful trick to simplify this calculation is to use current conservation for the
electron tensor L μν . For the electron transition current, the electromagnetic
current conservation condition is (cf equation (8.100))
q
μ [¯ u(k
′ , s
′ )γ μ u(k, s)] = 0
(8.188)
i.e. independent of the particular spin projections s and s
′ . Since L μν is
the product of two such currents, summed and averaged over polarizations,
current conservation implies the conditions
q
μ L μν = q
ν L μν = 0
(8.189)
which can be explicitly checked using our result for L μν . The usefulness of
′ in M
μν
this result is that in the contraction L μν M
μν we can replace p
by
(p + q) and then drop all the terms involving q’s, i.e.
μν
L μν M
μν =
(8.190)
L μν M eff
where
μν
μ
μν ].
M = 2[2p p
ν + (q
2 /2)g
(8.191)
eff
The calculation of the cross section is now straightforward. In the ‘laboratory’
system, defined (unrealistically) by the target muon at rest
p
μ = (M, 0, 0, 0)
(8.192)
with M now the muon mass, the result is (problem 8.17(a))
(
) (
)
2
dσ
dσ
q tan
2 (θ/2)
=
1 −
.
(8.193)
dΩ
dΩ
2M 2
ns
Note the following points:
Comment (a)
−
The ‘no-structure’ cross section (8.122) for e s
+ scattering now appears modified by an additional term proportional to tan
2 (θ/2). This is due to the spin1
2
nature of the muon which gives rise to scattering from both the charge and
the magnetic moment of the muon.
Comment (b)
In the kinematics the electron mass has been neglected, which is usually a
good approximation at high energies. We should add a word of explanation
for the ‘laboratory’ cross sections we have calculated, with the target muon
unrealistically at rest. The form of the cross section, (dσ/dΩ) ns , and of the
cross section for the scattering of two Dirac point particles, will be of great
value in our discussion of the quark parton model in the next chapter.
8. Elementary Processes in Scalar and Spinor Electrodynamics
To evaluate the cross section we must perform the ‘contraction’ L μν M
μν .
A useful trick to simplify this calculation is to use current conservation for the
electron tensor L μν . For the electron transition current, the electromagnetic
current conservation condition is (cf equation (8.100))
q
μ [¯ u(k
′ , s
′ )γ μ u(k, s)] = 0
(8.188)
i.e. independent of the particular spin projections s and s
′ . Since L μν is
the product of two such currents, summed and averaged over polarizations,
current conservation implies the conditions
q
μ L μν = q
ν L μν = 0
(8.189)
which can be explicitly checked using our result for L μν . The usefulness of
′ in M
μν
this result is that in the contraction L μν M
μν we can replace p
by
(p + q) and then drop all the terms involving q’s, i.e.
μν
L μν M
μν =
(8.190)
L μν M eff
where
μν
μ
μν ].
M = 2[2p p
ν + (q
2 /2)g
(8.191)
eff
The calculation of the cross section is now straightforward. In the ‘laboratory’
system, defined (unrealistically) by the target muon at rest
p
μ = (M, 0, 0, 0)
(8.192)
with M now the muon mass, the result is (problem 8.17(a))
(
) (
)
2
dσ
dσ
q tan
2 (θ/2)
=
1 −
.
(8.193)
dΩ
dΩ
2M 2
ns
Note the following points:
Comment (a)
−
The ‘no-structure’ cross section (8.122) for e s
+ scattering now appears modified by an additional term proportional to tan
2 (θ/2). This is due to the spin1
2
nature of the muon which gives rise to scattering from both the charge and
the magnetic moment of the muon.
Comment (b)
In the kinematics the electron mass has been neglected, which is usually a
good approximation at high energies. We should add a word of explanation
for the ‘laboratory’ cross sections we have calculated, with the target muon
unrealistically at rest. The form of the cross section, (dσ/dΩ) ns , and of the
cross section for the scattering of two Dirac point particles, will be of great
value in our discussion of the quark parton model in the next chapter.