Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 2 / a

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Solution

The first panel is the one-particle-irreducible Feynman diagram for photon vacuum polarization. The two internal Dirac propagators carry

k

and

k + q

, and the external photons carry

q

. The other panels fix the routings used in parts c and d.

Figure 1.
Original labelled one-loop QED Feynman diagrams: photon vacuum polarization, fermion self-energy, and the electromagnetic interaction vertex
. Details.

The closed fermion loop sign contributes a minus sign; there is no extra symmetry factor. Applying the given Feynman rules, with the external propagators amputated,

i Π_{μν} (q) = (- 1) \int_{k} tr [(- i e γ_{μ}) \frac{i ( \neq k + m )}{k ^{2} - m ^{2} + i 0} (- i e γ_{ν}) \frac{i ( \neq k + \neq q + m )}{( k + q ) ^{2} - m ^{2} + i 0}] .

Equivalently, the required amplitude is

Π_{μν} (q) = i e^{2} \int_{k} \frac{tr [ γ _{μ} ( \neq k + m ) γ _{ν} ( \neq k + \neq q + m )]}{( k ^{2} - m ^{2} + i 0 ) (( k + q ) ^{2} - m ^{2} + i 0 )} .

The Feynman slash notation is

\neq k = γ^{ρ} k_{ρ}

and

tr 1 = 4

in the chosen dimensional regularization convention.

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