230
8. Elementary Processes in Scalar and Spinor Electrodynamics
where u R and u L have positive and negative helicity respectively. The spinor
†
†
′
′
part of the matrix element (8.44) then becomes u R u R +u u L , from which it is
L
′
clear that helicity is conserved : the helicity of the u spinors equals that of the
′
u spinors; in particular there are no helicity mixing terms of the form u
† u L or
R
′
u
† u R . Consider then an initial state electron with positive helicity, and take
L
the z-axis to be along the incident momentum. The z-component of angular
momentum is then +
1 . Suppose the electron is scattered through an angle
2
of π. Since helicity is conserved, the scattered electron’s helicity will still be
positive, but since the direction of its momentum has been reversed, its angular
momentum along the original axis will be −
1 . Hence this configuration is
2
forbidden by angular momentum conservation – and similarly for an incoming
′
negative helicity state. The spin labels s , s in (8.46) can be taken to be
helicity labels and so it follows that the quantity S must vanish for θ = π in
the m → 0 limit. The ‘R’ and ‘L’ states are mixed by a mass term in the Dirac
equation (see (4.14) and (4.15)) and hence we expect backward scattering to
be increasingly allowed as m/E increases (recall that v = (1 − m
2 /E
2 )
1/2 so
that 1 − v
2 sin
2 θ/2 = cos
2 θ/2 + (m
2 /E
2 ) sin
2 θ/2).
8.2.2 Coulomb scattering of e − (field-theoretic approach)
Once again, the interaction Hamiltonian has been given in section 7.4, namely
ˆ ′
¯ ˆ
j
μ
H = −eψγ
μ ˆ
em,e
(8.51)
ψA μ ≡ ˆ
D
A μ
¯
j
μ
ˆ
where the current operator ˆ em,e is just −eψγ
μ ψ ˆ in this case. The lowest-order
amplitude is then
∫
′
ˆ ′
A e − = −i
′
| d
4 x H D (x)|e
− , k, s>
(8.52)
∫
′
= −i d
4 x
′
| ˆ j
μ
(x)|e
− , k, s>A μ (x).
(8.53)
em,e
With our normalization, and referring to the fermionic expansion (7.35), the
states are defined by
√
†
|e
− , k, s> = 2Ec ˆ (k)|0>
(8.54)
s
and similarly for the final state. We then find (problem 8.5) that the current
matrix element in (8.53) takes the form
′ ′
| ˆ j
μ
′ γ
μ
−i(k−k
' )·x
μ
− , k, s> = −eu ¯ ue
= j
− (x)
(8.55)
em,e
exactly as in (8.42). Thus once again, the ‘wavefunction’ and ‘field-theoretic’
approaches have been shown to be equivalent, in a simple case.
8.2.3 Trace techniques for spin summations
The calculation of cross sections involving fermions rapidly becomes laborious
following the ‘brute force’ method of section 8.2.1, in which the explicit forms
8. Elementary Processes in Scalar and Spinor Electrodynamics
where u R and u L have positive and negative helicity respectively. The spinor
†
†
′
′
part of the matrix element (8.44) then becomes u R u R +u u L , from which it is
L
′
clear that helicity is conserved : the helicity of the u spinors equals that of the
′
u spinors; in particular there are no helicity mixing terms of the form u
† u L or
R
′
u
† u R . Consider then an initial state electron with positive helicity, and take
L
the z-axis to be along the incident momentum. The z-component of angular
momentum is then +
1 . Suppose the electron is scattered through an angle
2
of π. Since helicity is conserved, the scattered electron’s helicity will still be
positive, but since the direction of its momentum has been reversed, its angular
momentum along the original axis will be −
1 . Hence this configuration is
2
forbidden by angular momentum conservation – and similarly for an incoming
′
negative helicity state. The spin labels s , s in (8.46) can be taken to be
helicity labels and so it follows that the quantity S must vanish for θ = π in
the m → 0 limit. The ‘R’ and ‘L’ states are mixed by a mass term in the Dirac
equation (see (4.14) and (4.15)) and hence we expect backward scattering to
be increasingly allowed as m/E increases (recall that v = (1 − m
2 /E
2 )
1/2 so
that 1 − v
2 sin
2 θ/2 = cos
2 θ/2 + (m
2 /E
2 ) sin
2 θ/2).
8.2.2 Coulomb scattering of e − (field-theoretic approach)
Once again, the interaction Hamiltonian has been given in section 7.4, namely
ˆ ′
¯ ˆ
j
μ
H = −eψγ
μ ˆ
em,e
(8.51)
ψA μ ≡ ˆ
D
A μ
¯
j
μ
ˆ
where the current operator ˆ em,e is just −eψγ
μ ψ ˆ in this case. The lowest-order
amplitude is then
∫
′
ˆ ′
A e − = −i
′
| d
4 x H D (x)|e
− , k, s>
(8.52)
∫
′
= −i d
4 x
′
| ˆ j
μ
(x)|e
− , k, s>A μ (x).
(8.53)
em,e
With our normalization, and referring to the fermionic expansion (7.35), the
states are defined by
√
†
|e
− , k, s> = 2Ec ˆ (k)|0>
(8.54)
s
and similarly for the final state. We then find (problem 8.5) that the current
matrix element in (8.53) takes the form
′ ′
| ˆ j
μ
′ γ
μ
−i(k−k
' )·x
μ
− , k, s> = −eu ¯ ue
= j
− (x)
(8.55)
em,e
exactly as in (8.42). Thus once again, the ‘wavefunction’ and ‘field-theoretic’
approaches have been shown to be equivalent, in a simple case.
8.2.3 Trace techniques for spin summations
The calculation of cross sections involving fermions rapidly becomes laborious
following the ‘brute force’ method of section 8.2.1, in which the explicit forms