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Past exam of the particle physics course of the University of Oxford / 2022 / 1 / b / Solution by

Codex  0 2026-10-25

There is one tree-level Feynman diagram: the distinguishable electron and muon exchange a photon in the

t

channel. There is no identical-particle exchange diagram.

Figure 1.
Electron-muon scattering by one-photon exchange
. **Details**.

Multiplying the external Dirac spinors, the two Feynman vertices and the photon propagator gives

i M = [\overset{u}{ˉ}_{3} (- i e γ^{μ}) u_{1}] \frac{- i g _{μν}}{q ^{2} + i ϵ} [\overset{u}{ˉ}_{4} (- i e γ^{ν}) u_{2}], M = \frac{e ^{2}}{q ^{2}} [\overset{u}{ˉ}_{3} γ^{μ} u_{1}] [\overset{u}{ˉ}_{4} γ_{μ} u_{2}] .

(1)

The overall phase of this scattering amplitude is convention dependent. Four external spin labels with two choices each give

2^{4} = 16

possible configurations before taking a limit. For

E ≫ m, M

, the vector QED interaction preserves fermion chirality, which becomes helicity for each external particle. Thus the surviving configurations, written as

(h_{e}, h_{μ})_{in} \to (h_{e}, h_{μ})_{out}

, are

LL \to LL, L R \to L R, R L \to R L, RR \to RR .

(2)

Here

L, R

specify helicity along each particle's own momentum. The other configurations require at least one mass-suppressed flip and vanish in the strictly massless limit.

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