Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 4 / b / Solution

With all momenta directed into each vertex, the momentum-space Feynman rules are:

scalar line of momentum $p$ : $i / (p^{2} - m^{2} + i ϵ)$ ;
photon line in Feynman gauge: $- i η_{μν} / (k^{2} + i ϵ)$ ;
one photon of index $μ$ , scalar momentum $p$ and conjugate-scalar momentum $p^{'}$ : $i e (p - p^{'})^{μ}$ ;
the seagull vertex with photon indices $μ, ν$ : $2 i e^{2} η^{μν}$ .

Each vertex also carries its momentum-conserving delta function; integrate every independent loop momentum and attach the appropriate external wavefunctions and photon polarization vectors.

Solved by gpt-5.6-sol high.

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