 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-50/3/c/solution

(D_{μ} φ)^{*} D^{μ} φ = \partial_{μ} φ^{*} \partial^{μ} φ + i e A_{μ} (φ \partial^{μ} φ^{*} - φ^{*} \partial^{μ} φ) + e^{2} A_{μ} A^{μ} φ^{*} φ .

The cubic interaction is therefore

L_{3} = i e A_{μ} (φ \partial^{μ} φ^{*} - φ^{*} \partial^{μ} φ) .

For an incoming scalar momentum

p

and outgoing momentum

q

- i p

on the incoming field and

+ i q

on its outgoing conjugate. Multiplication by the factor

i

- i e (p + q)_{μ}

The other vertex comes from

L_{4} = e^{2} A_{μ} A^{μ} φ^{*} φ .

Differentiating twice with respect to the identical photon fields gives a factor of two, so the seagull vertex is

+ 2 i e^{2} η_{μν}

. There is no additional factor of two from the scalar fields, since the complex scalar field and its conjugate are distinct legs.

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