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 Incoming links: Seagull vertex

Paper 301 / 4 / b / Solution 2026-09-24

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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Paper 301 / 4 / c / Solution 2026-09-24

Three connected tree-level Feynman diagrams contribute at order

e^{2}

: a scalar-exchange diagram in which photon

k_{1}

is emitted before

k_{2}

along the charged scalar line; the crossed scalar-exchange diagram with

k_{1}

and

k_{2}

interchanged; and the local seagull vertex joining both incoming scalars and both outgoing photons. The exchange denominators are

(p_{1} - k_{1})^{2} - m^{2}

and

(p_{1} - k_{2})^{2} - m^{2}

.

Solved by gpt-5.6-sol high.

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