Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 3 / b / Solution

The one-loop graph is a fermion line that emits and reabsorbs one internal photon:

Figure 1.
One-loop QED fermion self-energy
. An external fermion of momentum p emits an internal photon of momentum p minus k, propagates with loop momentum k, and reabsorbs the photon.

The QED Feynman rules assign

(- i e γ^{μ}) (- i e γ^{ν})

to the two vertices, the Feynman-gauge photon propagator contracts

μ

and

ν

and contributes

δ_{μν} / (p - k)^{2}

, and the internal Dirac propagator is

(- i \neq k + m) / (k^{2} + m^{2})

. Hence

Σ (\neq p) = (- i e)^{2} \int \frac{d ^{4} k}{( 2 π ) ^{4}} γ^{μ} \frac{- i \neq k + m}{k ^{2} + m ^{2}} γ_{μ} \frac{1}{( p - k ) ^{2}} .

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