250
8. Elementary Processes in Scalar and Spinor Electrodynamics
In the next chapter (section 9.5) we shall see how, in the latter process, meson
resonances dominate F (s).
The procedure whereby an ingoing/outgoing antiparticle is switched to
an outgoing/ingoing particle is called ‘crossing’ (the state is being ‘crossed’
from one side of the reaction to the other). By an extension of this language,
+ −
− π
+
e e → π
+ π
− is called the crossed process relative to e
− π
+
→ e
(or
vice versa). The fact that the amplitude for a given process and its ‘crossed’
analogue are directly related via the Feynman interpretation (or by quantum
field theory!) is called ‘crossing symmetry’. In the example studied here, what
is an s-channel process for one reaction becomes a t-channel process for the
crossed reaction. Essentially, little more is involved than looking in the one
case from left to right and, in the other, from top to bottom!
8.6 Electron Compton scattering
8.6.1 The lowest-order amplitudes
We proceed to explore some other elementary electromagnetic processes. So
far we have not considered a reaction with external photons, so let us now
discuss electron Compton scattering
′
γ(k, λ) + e
− (p, s) → γ(k
′ , λ
′ ) + e
− (p , s
′ )
(8.160)
where the λ’s stand for the polarizations of the photons. Since only the γ’s
′
and e
− ’s are involved, the interaction Hamiltonian is simply H ˆ , and it is
D
clear that this must act at least twice in the reaction (8.160). By following
the method of section 6.3.2 one can formally derive what we are here going to
assume is by now obvious, which is that to order e
2 (i.e. α in the amplitude)
there are two contributing Feynman graphs, as shown in figures 8.14(a) and
(b). The first is an s-channel process, the second a u-channel process. We
already know the factors for the vertices and for the external electron lines; we
need to know the factors for the internal electron lines (propagators) and the
external photon lines. The fermion propagator was given in section 7.2 and is
i/( / q − m + i∈) for a line carrying 4-momentum q. As regards the ‘external-γ’
factor, this will arise from contractions of the form (cf (6.90))
√
2E k ' <0|α(k
′ , λ
′ )A ˆ μ (x 1 )|0> = ∈
μ∗ (k
′ , λ
′ )e
ik
' ·x1
(8.161)
where the evaluation of the vev has used the mode expansion (7.104) and the
commutation relations (7.108), as usual; note, however, that only transverse
′
polarization states (λ, λ = 1 and 2) enter in the external (physical) photon
lines in figures 8.14(a) and (b).
8. Elementary Processes in Scalar and Spinor Electrodynamics
In the next chapter (section 9.5) we shall see how, in the latter process, meson
resonances dominate F (s).
The procedure whereby an ingoing/outgoing antiparticle is switched to
an outgoing/ingoing particle is called ‘crossing’ (the state is being ‘crossed’
from one side of the reaction to the other). By an extension of this language,
+ −
− π
+
e e → π
+ π
− is called the crossed process relative to e
− π
+
→ e
(or
vice versa). The fact that the amplitude for a given process and its ‘crossed’
analogue are directly related via the Feynman interpretation (or by quantum
field theory!) is called ‘crossing symmetry’. In the example studied here, what
is an s-channel process for one reaction becomes a t-channel process for the
crossed reaction. Essentially, little more is involved than looking in the one
case from left to right and, in the other, from top to bottom!
8.6 Electron Compton scattering
8.6.1 The lowest-order amplitudes
We proceed to explore some other elementary electromagnetic processes. So
far we have not considered a reaction with external photons, so let us now
discuss electron Compton scattering
′
γ(k, λ) + e
− (p, s) → γ(k
′ , λ
′ ) + e
− (p , s
′ )
(8.160)
where the λ’s stand for the polarizations of the photons. Since only the γ’s
′
and e
− ’s are involved, the interaction Hamiltonian is simply H ˆ , and it is
D
clear that this must act at least twice in the reaction (8.160). By following
the method of section 6.3.2 one can formally derive what we are here going to
assume is by now obvious, which is that to order e
2 (i.e. α in the amplitude)
there are two contributing Feynman graphs, as shown in figures 8.14(a) and
(b). The first is an s-channel process, the second a u-channel process. We
already know the factors for the vertices and for the external electron lines; we
need to know the factors for the internal electron lines (propagators) and the
external photon lines. The fermion propagator was given in section 7.2 and is
i/( / q − m + i∈) for a line carrying 4-momentum q. As regards the ‘external-γ’
factor, this will arise from contractions of the form (cf (6.90))
√
2E k ' <0|α(k
′ , λ
′ )A ˆ μ (x 1 )|0> = ∈
μ∗ (k
′ , λ
′ )e
ik
' ·x1
(8.161)
where the evaluation of the vev has used the mode expansion (7.104) and the
commutation relations (7.108), as usual; note, however, that only transverse
′
polarization states (λ, λ = 1 and 2) enter in the external (physical) photon
lines in figures 8.14(a) and (b).