234
8. Elementary Processes in Scalar and Spinor Electrodynamics
since the ‘v’ solutions have been set up precisely to correspond to the ‘−k, −s’
situation. In evaluating the cross section from (8.83), the only difference from
the e
− case is the appearance of the spinors ‘v’ rather than ‘u’; the lepton
tensor in this case is
′
L
μν
1
= Tr[(k / − m)γ
μ (k / − m)γ
ν ]
(8.84)
2
∑
using the result (7.64) for
v(k, s)¯ v(k, s). Expression (8.84) differs from
s
(8.67) by the sign of m and by k ↔ k
′ , but the result (8.79) for the trace
is insensitive to these changes. Thus the positron Coulomb scattering cross
section is equal to the electron one to lowest order in α.
′
In the field-theoretic approach, the same interaction Hamiltonian H ˆ
D
which we used for e
− scattering will again automatically yield the e
+ matrix element (recall the discussion at the end of section 8.1.3). In place of
(8.53), the amplitude we wish to calculate is
∫
′
A e + = −i d
4 x
′
| ˆ j
μ
(x)|e
+ , k, s>A μ (x)
em,e
∫
′
¯
= −i d
4 x
′
| − eψ ˆ (x)γ
μ ψ ˆ (x)|e
+ , k, s>A μ (x) (8.85)
where, referring to the fermionic expansion (7.35),
√
|e
+ , k, s> = 2Ed
† (k)|0>,
(8.86)
s
and similarly for the final state. In evaluating the matrix element in (8.85) we
must again remember to normally order the fields, according to the discussion
in section 7.2. Bearing this in mind, and inserting the expansion (7.35), one
finds (problem 8.9)
′
′
)·x
′
| ˆ j
μ
(x)|e
+ , k, s> = +ev ¯(k, s)γ
μ v(k , s
′ )e
−i(k−k
'
(8.87)
em,e
≡ j
μ
(x)
(8.88)
em,e +
just as required in (8.83). Note especially that the correct sign has emerged
naturally without having to be put in ‘by hand’, as was necessary in the
wavefunction approach when applied to an antifermion.
We are now ready to look at some more realistic (and covariant) processes.
8.3 e
− s
+ scattering
+
− +
8.3.1 The amplitude for e − s → e s
We consider the two-body scattering process
′
e
− (k, s) + s
+ (p) → e
− (k , s
′ ) + s
+ (p
′ )
(8.89)
8. Elementary Processes in Scalar and Spinor Electrodynamics
since the ‘v’ solutions have been set up precisely to correspond to the ‘−k, −s’
situation. In evaluating the cross section from (8.83), the only difference from
the e
− case is the appearance of the spinors ‘v’ rather than ‘u’; the lepton
tensor in this case is
′
L
μν
1
= Tr[(k / − m)γ
μ (k / − m)γ
ν ]
(8.84)
2
∑
using the result (7.64) for
v(k, s)¯ v(k, s). Expression (8.84) differs from
s
(8.67) by the sign of m and by k ↔ k
′ , but the result (8.79) for the trace
is insensitive to these changes. Thus the positron Coulomb scattering cross
section is equal to the electron one to lowest order in α.
′
In the field-theoretic approach, the same interaction Hamiltonian H ˆ
D
which we used for e
− scattering will again automatically yield the e
+ matrix element (recall the discussion at the end of section 8.1.3). In place of
(8.53), the amplitude we wish to calculate is
∫
′
A e + = −i d
4 x
′
| ˆ j
μ
(x)|e
+ , k, s>A μ (x)
em,e
∫
′
¯
= −i d
4 x
′
| − eψ ˆ (x)γ
μ ψ ˆ (x)|e
+ , k, s>A μ (x) (8.85)
where, referring to the fermionic expansion (7.35),
√
|e
+ , k, s> = 2Ed
† (k)|0>,
(8.86)
s
and similarly for the final state. In evaluating the matrix element in (8.85) we
must again remember to normally order the fields, according to the discussion
in section 7.2. Bearing this in mind, and inserting the expansion (7.35), one
finds (problem 8.9)
′
′
)·x
′
| ˆ j
μ
(x)|e
+ , k, s> = +ev ¯(k, s)γ
μ v(k , s
′ )e
−i(k−k
'
(8.87)
em,e
≡ j
μ
(x)
(8.88)
em,e +
just as required in (8.83). Note especially that the correct sign has emerged
naturally without having to be put in ‘by hand’, as was necessary in the
wavefunction approach when applied to an antifermion.
We are now ready to look at some more realistic (and covariant) processes.
8.3 e
− s
+ scattering
+
− +
8.3.1 The amplitude for e − s → e s
We consider the two-body scattering process
′
e
− (k, s) + s
+ (p) → e
− (k , s
′ ) + s
+ (p
′ )
(8.89)