 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2021/1/b/ii/solution

For

Ψ (p_{1}) + ϕ (k_{1}) \to Ψ (p_{2}) + ϕ (k_{2})

, there are two tree-level Feynman diagrams, both with an internal

χ

fermion. Their internal four-momenta are

p_{1} + k_{1}

and

p_{1} - k_{2}

i M = (- i g)^{2} \overset{u}{ˉ}_{Ψ} (p_{2}) [S_{χ} (p_{1} + k_{1}) + S_{χ} (p_{1} - k_{2})] u_{Ψ} (p_{1}) .

These are the two possible absorption/emission orderings along the fermion flow. There is no third tree-level Feynman diagram using the cubic scalar field interaction vertex, since attaching its remaining scalar field line to the external

Ψ

line would require a forbidden

ϕ \overset{ˉ}{Ψ} Ψ

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